NCERT Solutions Class 5 Maths Chapter 7 Shapes and Patterns

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 5 · M AT H S

NCERT Solutions

Chapter 7: Shapes and Patterns

NCERT Textbook — Math Mela

BOOK PAGES SECTIONS QUESTIONS MEDIUM

92 – 103 18 63 English

Solutions, notes, sample papers & more at 90 pages

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

CLASS 5 · MATHS · MATH MELA

NCERT Solutions — Chapter 7: Shapes and Patterns
Chapter 7 of Math Mela is a workshop, not a lesson. You weave a paper mat, fit cut-out shapes around a point,
cut a rhombus into four triangles and build new shapes with them, play with a tangram, stitch a design on a
circle, count small cubes inside a big one, and fold paper nets into an icosahedron and a dodecahedron.
Along the way you meet tessellation, the isosceles, equilateral and scalene triangles, the kite, the
parallelogram and the rectangle.

TEXTBOOK BOOK PAGES

Math Mela (Class 5) 92 – 103

SECTIONS QUESTIONS

18 63

MEDIUM

English

Weaving Mats — Pages 92–93

Page 1 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Making paper mats and writing the rule for each row

WEAVING MATS

Q1 Let us make paper mats. You will need — A coloured paper (30 cm long and 20 cm
wide) and eight paper strips of two different colours (3 cm wide and longer than 20
cm). (a) Take a coloured paper 30 cm long and 20 cm wide. (b) Fold the coloured
paper in half along the longer side. (c) Draw vertical lines at equal distances from
the closed end and cut slits leaving a gap of 3 cm at the top. (d) Carefully unfold the
paper. There will be no cuts in the paper at the top and the bottom. (e) Now cut 8
paper strips of 3 cm width in 2 colours and of length slightly longer than 20 cm. (f)
Take one colour strip and weave it across the slits going 1 under and 1 over, and
again 1 under and 1 over. Repeat it for the first row. (g) Take one more strip of
another colour and weave it across the slits going 1 over and 1 under, and again 1
over and 1 under. Repeat it for the second row. (h) Weave all the strips in the same
alternating pattern. Neatly fold any extra strip ends behind the mat. Your mat is
ready!

Step (b) Step (c) Step (f)

Step (h)

Steps (b), (c), (f) and (h) of making the paper mat.

Follow the eight steps and you get a chequered mat — it looks like a chessboard, with the
two colours swapping places in every row.
Here is the same activity written as short steps you can do at your desk.

1. Take the base paper. 30 cm long, 20 cm wide.
2. Fold it in half along the longer side. Now it is 15 cm by 20 cm and one side is a closed fold.

Page 2 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

3. Draw lines and cut. From the folded (closed) edge, draw straight lines the same distance
apart. Cut along them but stop 3 cm before the open edge. Do not cut all the way.
4. Open the paper. You now have a row of long slits in the middle, with 3 cm of uncut paper at
the top and at the bottom. These uncut bands hold the mat together.
5. Cut 8 strips, 3 cm wide, a little longer than 20 cm — 4 of one colour and 4 of the other.
6. Weave the first row. Take a strip. Push it under the first paper band, over the next, under,
over … all the way across. Slide it up to the top.
7. Weave the second row with the other colour, but start the opposite way: over, under, over,
under …
8. Keep alternating. Row 3 is like Row 1, Row 4 is like Row 2, and so on. Fold the extra ends
behind and press the mat flat.

Row 1 — 1 under, 1 over, 1 under, 1 over, … (repeat)

Row 2 — 1 over, 1 under, 1 over, 1 under, … (repeat)

Row 3 — same as Row 1 Row 4 — same as Row 2

Row 1 starts under, Row 2 starts over — a
chessboard is born

The cream squares are the base paper showing through; the coloured squares are the strips lying on
top.

Why it happens: In Row 1 the strip hides band 1, shows band 2, hides band 3 … In
Row 2 it does exactly the opposite. So no coloured square ever sits directly below
another of the same kind, and the mat looks like a chessboard.

Page 3 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
m.
Tip: Push each new strip right up against the one before it. If you leave a gap, the

m as e
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pattern still works but the mat looks loose. Fold the ends behind and put a little gum
on them. .co a g
a s em
agl

co m
ag
Can you figure out how to make this mat? Let us try to understand how this mat is
.
Q2

e m
as
woven by looking at the pattern in the first two rows. Row 1 — 2 over, 1 under, 2

a g l
over, 1 under, … Row 2 — 2 under, 1 over, 2 under, 1 over, … You can use strips of the
same colour or 2 different colours, one for each row.

co m
e m.
m l as
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l a se
a g
m a s
m .co agl
l a se
a g

co m
The mat shown in the book, page 93.
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m as e
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a s em
agl ANSWER

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Yes. Weave Row 1 as 2 over, 1 under again and again, and Row 2 as 2 under, 1 over again and

a se
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again. Then Row 3 is like Row 1, Row 4 is like Row 2, and so on.
. a g
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How to read the words while you weave:

agl
1. “2 over” means: pass the strip over the first band and over the second band.
2. “1 under” means: now push it under the next band.

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.
3. Repeat — over, over, under, over, over, under … till the end of the row.

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4. Row 2 is the reverse. Under, under, over, under, under, over …
m l
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m Row 1 — O O U O O U O O U … a g
l a se
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Row 2 — U U O U U O U U O …

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(O = strip goes over, U = strip goes under)
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m ase
.co


a g l Page 4 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Row 1 above, Row 2 below — one is the mirror
of the other

over over under over over under

under under over under under over

Blue = strip on top Cream = base paper showing
Each row hides two bands, then shows one. The next row does exactly the opposite.

Why it happens: The rule has three steps in it (over, over, under), so the picture
repeats after every 3 bands. Row 2 turns each “over” into an “under”, so the long
coloured blocks of Row 1 sit above the gaps of Row 2. That is what makes the brick-
like look of the mat.

Two colours or one? Both work. With one colour you see the pattern only from the
light and shade. With two colours the blocks jump out. Use one colour for the odd
rows and the other for the even rows.

Q3 Try to weave a pattern, using the rules given below. Row 1 — 2 over, 1 under, 2 over,
1 under, … (repeat). Row 2 — 1 under (do not repeat), 3 over, 3 under, 3 over, 3 under,
… (repeat). Row 3 — 2 under, 1 over, 2 under, … (repeat). Row 4 — 1 over (do not
repeat), 3 under, 3 over … (repeat). Continue weaving in this order.

Here is the pattern written out band by band, for a mat with 12 bands. Weave your strips
exactly like this.

Page 5 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

BAND NUMBER 1 2 3 4 5 6 7 8 9 10 11 12

Row 1 O O U O O U O O U O O U

Row 2 U O O O U U U O O O U U

Row 3 U U O U U O U U O U U O

Row 4 O U U U O O O U U U O O

(O = the strip passes over that band · U = the strip passes under it.)
Read each rule like this:

1. Row 1: over, over, under — then start the same three again.
2. Row 2: the very first band is under, and that first step is not repeated. After it, go 3 over, 3
under, 3 over, 3 under …
3. Row 3: under, under, over — then start again.
4. Row 4: the very first band is over, not repeated. After it, 3 under, 3 over, 3 under …
5. Row 5 is the same as Row 1, Row 6 the same as Row 2, and so on. The whole design repeats
after every 4 rows.

The coloured blocks are the parts of the strip you
can see

R1

R2

R3

R4

After Row 4 the whole set of four rules starts again.
Four rows, four different rules. Together they make one repeating band of the design.

Tip while weaving: Say the rule aloud — “over, over, under” — as your fingers move.
If you lose your place, count back to the last group of three and start again from
there.

Page 6 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q4 Can you work out the steps for any of these designs and weave the pattern? Write
the steps of the pattern in your notebook for each row until it starts repeating.

The two woven designs given in the book, page 93.

Yes. Here is the method, and then a worked example for the first of the two mats shown (the
purple and pale-blue one).
Method — how to read any woven design

1. Look at the top row only. Put your finger on the left end.
2. Move along the row. Each time the crossing strip is on top write O (over); each time the
upright band is on top write U (under).
3. Write the row as words: for example O O O U becomes “3 over, 1 under”.
4. Do the same for the next row, and the next.
5. Stop when a row is exactly the same as an earlier row. Everything between them is the
repeating block.

Sample answer — the purple mat

Page 7 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Row 1 — 3 over, 1 under, … (repeat)

Row 2 — same, but shifted one band to the right

Row 3 — shifted one more band
Row 4 — shifted one more band

Row 5 — same as Row 1, so the design repeats after 4 rows

3 over, 1 under — and every row slides one step to
the right

The sliding is what makes the slanting lines you can
see in the mat.

Purple = the crossing strip on top. Sliding the rule one band each row turns a straight pattern into a
slanting one.

Why the design slants: Each row does the same thing as the row above it, but one
band later. So the long purple blocks step sideways, one place at a time, and your
eye joins them into a diagonal stripe.

Try this yourself: Take the second mat and write its rows the same way. Then invent
your own — for example “2 over, 2 under” shifted by two bands each row — and
weave it. Any rule you can write down, you can weave.

Let Us Try — Page 93

Page 8 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

Completing the grid design
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LET US TRY

.co a g l
a s em the following pattern on a grid paper. Part of it is done for you. Now, complete
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Draw
a
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the rest of the grid to get the full design.

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Left — the design to be drawn. Right — the dotted grid, with part of it already done.

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m as e
.co a g l
a s emANSWER
agl Finish the shape by drawing its mirror image below. The part already drawn is the top half of
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a big diamond. Add the bottom half and the design is complete.
s e m
m a
Look carefully at what is already there:
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g l as
a
The top row has shaded squares with one empty square between them — shade one, skip
one, shade one, skip one.

co m
m .
m ase
.co


a g l Page 9 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Below it are two lines of shaded squares that slope down from the middle, one going left and
one going right. Together they make a big Λ shape, like the roof of a house.

Steps to complete it

1. Finish the top row — keep shading alternate squares right to the edge of the grid.
2. Find the lowest shaded square on the left slope and on the right slope.
3. Turn the Λ upside down. From those two lowest squares, draw two new slopes going down
and inwards, until they meet in the middle. Now the two slopes make a large diamond.
4. Shade a small solid diamond in the middle of the big one, as in the woven mat picture.
5. Repeat the diamond across the grid if there is room, and colour it.

Yellow = already drawn in the book · Green =
the part you add

The two halves together make one big diamond.
Add the lower “V” to the upper “Λ” and a diamond appears, with a small solid diamond at its heart.

Why it works: The design has a mirror line straight across its middle. Whatever is
above the line is repeated below it, turned upside down. Once you spot the mirror
line, the rest of the drawing is only copying.

Page 10 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Check it yourself: Fold your finished drawing along the middle line. The two halves
should land exactly on top of each other. If a square sticks out, you have shaded one
square too many.

Tiling and Tessellation — Page 94
Three regular pentagons around a point

TILING AND TESSELLATION

Q1 We have placed 3 pentagons around a point. Can we fit one more into the empty
space?

No, a fourth pentagon will not fit. The empty space left is far too thin for it.

Page 11 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Three pentagon corners meet. A thin
gap is left over.

gap

108 + 108 + 108 = 324. A full turn is
360. So the gap is only 36.
Three corners of a regular pentagon do not quite fill the turn around a point, and what is left is much
too small for a fourth pentagon.

Corner of a regular pentagon = 108

Three corners = 108 + 108 + 108 = 324

A full turn round a point = 360

Space left over = 360 − 324 = 36

But one pentagon corner needs 108, and 108 is much more than 36.

Page 12 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

So there are only two choices, and both are wrong:

1. Leave the gap — then the floor has a hole in it, which is not allowed in tiling.
2. Push a fourth pentagon in — then it climbs on top of the others. That is an overlap, which
is also not allowed.

Why it happens: To cover a floor with no gaps, the corners that meet at a point
must add up to exactly one full turn. For pentagons the corner size 108 does not
divide 360 evenly — 3 corners are too few and 4 are too many. So regular
pentagons do not tessellate.

Try it yourself: Cut out five or six pentagons from the sheet at the end of the book
and lay them round a dot on your desk. You will see the same thin gap every single
time, no matter how you turn them.

Find Out — Page 94
Which regular shapes fit around a point

FIND OUT

Q1 Can regular triangles fit together at a point without any gap? How many of them fit
together? (A sample triangle is given at the end of the book.)

Yes, they fit perfectly. Six triangles fit around one point.

Page 13 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
Six oequal triangles close up exactly se— m.
m g l a
m .c a
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asand make a hexagon
agl

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60 + 60 + 60 + 60 + 60 + 60 = 360 l a se
m .c ag
l a se
ag Six corners of 60 make one full turn. No gap is left and no triangle sits on another.

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Corner of an equilateral triangle = 60.co
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How many make a full turn?l360 a se÷m60 = 6
ag
Check: 60 × 6 = 360 ✓

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Why it happens: A triangle with all three sides equal also has all three angles equal,

s e m and each of them is 60. Six of these corners together make exactly one complete
agla turn, so the triangles close up with nothing left over. That is why equilateral
.c
triangles tessellate.
s e m
m a
m . co agl
l a se
Did you notice? The six triangles together make a regular hexagon. That is why
ag tiles look so good side by side.
hexagon tiles and triangle

com
m .
m ase
.co


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Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q2 Can squares (a regular 4-sided shape) fit together around a point without any gap
or overlap? Try it out using cutouts of squares (a sample square is given at the end
of the book). How many squares did you need?

Yes, squares fit together with no gap and no overlap. You need 4 squares.

Four right angles meet at the red dot

90 + 90 + 90 + 90 = 360 — one full
turn
This is why square floor tiles cover a
room so neatly.

Four square corners fit round a point exactly. Nothing is left over and nothing overlaps.

Page 15 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Corner of a square = a right angle = 90

360 ÷ 90 = 4 squares

Check: 90 × 4 = 360 ✓

Why it happens: Every corner of a square is a right angle. Four right angles make a
complete turn — think of turning a quarter turn four times to face the front again.
So four squares close up exactly, and squares tessellate.

Look around you: The tiles on your classroom floor, a chess board, the squares on
graph paper — all of them are squares fitting together four at a corner.

Q3 Can five squares fit together around a point without any gaps or overlaps? Why or
why not?

No. Five squares cannot fit around a point. The fifth one has to climb on top of the others.

Four squares already fill the turn: 90 × 4 = 360

Five squares would need: 90 × 5 = 450

450 is bigger than 360 by 90

So one whole right angle has nowhere to go.

Page 16 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

The four blue squares already use up
the whole turn

The fifth (red) square can only lie on top
— that is an overlap.

There is no room left for a fifth square, so it overlaps the others.

Why it happens: Around any point there is exactly one full turn to share out, and
that is 360. Four square corners of 90 use it all up. A fifth corner needs another 90,
and there is simply none left.

The same rule everywhere: Shapes fit round a point only when their corners add
up to exactly 360 — not less (gap) and not more (overlap).

Q4 Can regular hexagons (6-sided shapes with equal sides) fit together around a point
without any gaps or overlaps? Try and see (a sample hexagon is given at the end of
the book). How many fit together at a point?

Yes. Three regular hexagons fit around a point.

Page 17 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Three hexagon corners close the turn
exactly

120 + 120 + 120 = 360
Three regular hexagons meet at every point — the same shape a honeycomb makes.

Corner of a regular hexagon = 120

360 ÷ 120 = 3 hexagons

Check: 120 × 3 = 360 ✓

Why it happens: A regular hexagon can be cut into 6 equal triangles from its centre,
and each corner of the hexagon is made of two 60 angles. That makes 120 at each
corner, and three of them make a full turn. So regular hexagons tessellate.

Did you know? Honeybees build their comb out of hexagons. Hexagons hold the
most honey for the least wax, and they leave no gaps at all.

Page 18 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
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Tiling and Tessellation — Page 95
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A pattern with more than one shape, and the regular octagon
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TILING AND TESSELLATION

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Here is a tessellating pattern with more than one shape. What shapes have been
m
Q1
used in this pattern? _____________, _____________.
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The tiling pattern given in the book, page 95.

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a Regular hexagons and equilateral triangles.

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Look at the picture in the book. The big orange shapes have six equal sides — those are regular
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hexagons. The blue shapes between them have three equal sides — those are equilateral

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triangles.

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a g l Page 19 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Orange hexagons with blue triangles filling every gap. No holes, no overlaps.

At every meeting point: 2 hexagon corners + 2 triangle corners

= 120 + 120 + 60 + 60

= 360 — exactly one full turn ✓

Why two shapes are used: Hexagons alone tessellate, and triangles alone
tessellate. Here they are mixed, and the mix still works because the corners that
meet still add up to 360. A tiling may use more than one shape as long as that rule is
kept.

Look for it: This pattern is common on rangoli designs, dupatta prints and old tile
floors. Once you know the two shapes, you can spot it everywhere.

Page 20 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q2 Continue the pattern given below and colour it appropriately.

The part of the pattern printed in the book, page 95.

Keep adding a purple hexagon with six yellow triangles around it, again and again. The
little piece drawn in the book is the start of a hexagon-and-triangle tiling.
Steps

1. Look at what is drawn. Each purple hexagon has yellow triangles sitting on its sides.
2. Pick a free side of a hexagon. Put a yellow triangle on it, with its side exactly along the
hexagon's side.
3. Where two triangles point at each other, draw a new purple hexagon in the space
between them.
4. Repeat until the dotted box is full.
5. Colour all the hexagons in one colour and all the triangles in another. Same shape, same
colour.

Page 21 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

The finished pattern: purple hexagons, yellow triangles, no gaps anywhere.

Why it never leaves a gap: At every corner two hexagons and two triangles meet.
120 + 120 + 60 + 60 = 360, one full turn. Because the rule is the same at every corner,
the pattern can go on for ever.

Colouring tip: Colour by shape, not by position. All hexagons one colour, all
triangles another. Then the pattern shows up clearly from across the room.

Q3 A regular octagon means a shape with eight equal sides. Do regular octagons fit
together without any gaps or overlaps? Try drawing the same and check.

No. Regular octagons do not tessellate. Wherever two of them meet, a square hole is left
behind.

Page 22 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Two octagons meet — and a
square-shaped hole appears

gap

135 + 135 = 270. The turn is 360, so 90
is left empty.
Two octagon corners do not fill the turn, and a third octagon is far too big for what is left.

Corner of a regular octagon = 135

Two corners = 135 + 135 = 270, gap left = 360 − 270 = 90

Three corners = 135 × 3 = 405, which is more than 360 → they would overlap

So neither 2 nor 3 octagons work.

Page 23 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
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Why it happens: 360 divided by 135 is not a whole number. Two octagons leave a

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com always has holes. l
gap of 90, and a third cannot squeeze into 90 because it needs 135. So a floor of
octagons.alone a g
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agl
But there is a trick: The leftover gap is exactly 90 — the corner of a square. So if

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you use octagons and small squares together, the floor is covered perfectly. Many
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old verandah floors are tiled exactly like
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Tiling and Tessellation — Page 96
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What meets at a marked point; squares and triangles together

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TILING AND TESSELLATION

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Look at the pattern given below. What shapes are coming together at the marked
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points? Are the same set of shapes coming together at these points? Continue the
agl
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pattern and colour it appropriately.
a
agl

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empoints are shown.
The dotted grid on page 96, with the four marked points. In the printed book the

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pattern itself is missing from this grid — only the marked

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m a s e
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First, a note about the book. In the printed page the top dotted box is empty — only the four

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red dots are shown, and the tiles themselves have not been printed. So there is nothing there to

l a
ag below on any tiling.
read off. Use the second pattern on the same page (the one with squares and triangles) for this
question, and use the method

m
How to answer “what is coming together at a point”

. co
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m l as
.co a g
Page 24 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

1. Put your pencil tip on the marked point.
2. Go round it once, like the hand of a clock. Name each tile you pass over.
3. Write the list in order — for example: square, triangle, square, triangle, triangle.
4. Do the same at the next marked point and compare the two lists.
5. If both lists have the same shapes in the same order, the answer is “yes, the same set of
shapes comes together at every point”.

Check with the corner sizes:

square corner = 90 · equilateral-triangle corner = 60

3 triangles + 2 squares = 60 + 60 + 60 + 90 + 90 = 360 ✓

Sample answer: At each marked point three equilateral triangles and two squares come
together. The same five shapes meet at every marked point, so the pattern is the same all over.
To continue it, keep placing squares and triangles so that this same group of five appears at
every new corner, then colour all the squares one colour and all the triangles another.

If your copy of the book does show a pattern in that box, use the same five steps
on it: walk round each red dot, list the shapes, and compare.

Page 25 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q2 Here is a tiling pattern made using two different shapes — squares and triangles.
Are the triangles equilateral? Why or why not?

The tiling pattern on page 96, with the marked points in red.

Yes, the triangles are equilateral — all three of their sides are equal.
Here is the reason, in three short steps.

1. Look at any triangle in the pattern. Each of its sides is also the side of a square that
touches it.
2. All four sides of a square are equal. So every side of the triangle is the same length as a
side of a square.
3. All the squares in the pattern are the same size. So all three sides of the triangle are equal
— that is exactly what “equilateral” means.

Page 26 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

At the big red dot: 2 squares + 3
triangles

The dotted triangle is the one that fills the
last gap.
Every side of every orange triangle is also the side of a purple square, so all the triangle's sides are
equal.

Why the pattern works: At the big red dot, two squares and three triangles meet.
90 + 90 + 60 + 60 + 60 = 360, one full turn. Because the corners add up exactly, the
tiles cover the floor with no gap and no overlap.

Page 27 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Check it yourself: Measure any triangle side and any square side in the picture with
your ruler. They come out the same. That is the proof, done with a ruler instead of
words.

Q3 What shapes are coming together at the marked points? Are the same set of shapes
coming together at these points? Continue the pattern and colour it appropriately.

The tiling pattern on page 96, with the marked points in red.

Three equilateral triangles and two squares come together at each marked point. Yes —
the same five shapes meet at every marked point.
Walk round the big red dot in the picture above and count what you pass:

Page 28 of 90

as e
Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
m.
GOING ROUND THE POINT SHAPE CORNER SIZE

m as e
1st
.co
Square
a g l90

se m
g l a
a
2nd Triangle 60

3rd Square 90

co m
4th
e m .Triangle 60 ag
g l as
5th
a Triangle 60

Total 2 squares + 3 triangles 360
co m
e m.
m l as
.co g
How to continue the pattern

em a
a s
1. Find a corner that is not yet full. Count the shapes already round it.

a gl2. Work out what is missing. Each corner needs 2 squares and 3 triangles in all.
3. Draw the missing tile with its side exactly along a side that is already there.

om a s
. c agl
4. Move to the next unfinished corner and do the same.

s e mtriangles another.
5. Colour last: all squares one colour, all

a gla
Why the same set appears everywhere: The tiling was built by repeating one small
group of tiles over and over. Repeating a group means every point ends up looking
co m
m .
e
like every other point — that is what makes it a pattern and not just a jumble.
m l as
.co
m Work outwards from the middle, one ring at a time. If you jump about, you will a g
a s eTip:
a gl leave holes that are hard to fill later.

se m
com g l a
m . a
Create similar patternsla
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s other cutouts of shapes.
Q4
a g using

. c om
Use the cut-outs from the sheet at the end of your book — squares, e
s m
m a
equilateral triangles,

. co hexagons and regular pentagons.
regular
a gl
asem
l
Steps
a g 1. Choose your shapes. Start with two kinds, for example hexagons and triangles.
.c
s e m
m a
co agl
2. Put one shape down in the middle of a sheet of paper.

m .
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3. Add shapes side by side, always matching a whole side to a whole side. Never let one cut-
out climb on another.
l a
g corners meeting there must add up to 360 — no gap, no overlap.
4. Check every corner.aThe
5. Trace round the cut-outs with a pencil, lift them off, and colour your pattern.

co m
m .
m as e
.co


a g l Page 29 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Sample answer — three patterns that work

PATTERN SHAPES USED WHAT MEETS AT EACH POINT CHECK

1 Squares only 4 squares 90 × 4 = 360

2 Triangles only 6 triangles 60 × 6 = 360

3 Hexagons and triangles 2 hexagons + 2 triangles 120 + 120 + 60 + 60 = 360

4 Squares and triangles 2 squares + 3 triangles 90 + 90 + 60 + 60 + 60 = 360

One that will not work: Pentagons on their own. Try them and watch the thin gap
appear every time. Finding out what does not work is just as useful as finding what
does.

Shapes from the Rhombus — Pages 96–97
Cutting a rhombus into four triangles and building with them

Q1 You will find a copy of this rhombus at the end of the book. Cut out the triangular
pieces for the following activities. What geometrical shapes can you make by fitting
2 of these triangles together? Trace the shapes you created.

The rhombus on page 96, divided into four triangles.

With 2 pieces you can make five different shapes: two kinds of triangle, a kite, a
parallelogram and a rectangle.

Page 30 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

First, look at one piece. The rhombus was cut along both its long lines (the diagonals), so all four
pieces are the same. Each piece is a right-angled triangle: it has one square corner, one short
side, one long side and one slanting side.

One rhombus, two cuts, four
equal pieces

1 2

3 4

Every piece has a square corner in
the middle.
The two diagonals cut the rhombus into 4 equal right-angled triangles.

Now join two pieces, always side to side.

Page 31 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

JOIN THEM ALONG… HOW SHAPE YOU GET SIDES

the long side as a mirror A tall isosceles triangle 3

the short side as a mirror A wide isosceles triangle 3

the slanting side as a mirror A kite 4

the slanting side turned half a turn A rectangle 4

a short or long side turned half a turn A parallelogram 4

triangle triangle kite rectangle
parallelogra

Five shapes from the same two pieces. The dotted line shows where they are joined.

Trace them: Lay the two pieces down, draw round the outside with a pencil, then lift
them off. Write the name of the shape inside your tracing.

Q2 How many different types of triangles can you make? Now, observe and measure
the sides of these triangles. What do you notice?

You can make 2 different types of triangle. Both of them have exactly 2 equal sides — they
are isosceles triangles.
What you see when you measure

1. Triangle 1 — the tall one. Join two pieces along their long sides. Its base is the short line of
the rhombus. Its other two sides are two sides of the rhombus, so they are equal.
2. Triangle 2 — the wide one. Join two pieces along their short sides. Its base is the long line
of the rhombus. Again the other two sides are sides of the rhombus, so they are equal.

Page 32 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

3. Measure with your ruler. In both triangles the two slanting sides come out the same, and
the base is different from them.

Triangle 1: side, side, short diagonal → two sides equal

Triangle 2: side, side, long diagonal → two sides equal

A triangle with exactly two equal sides is called an isosceles triangle.

Why the two sides are always equal: A rhombus has all four sides the same length.
When you cut it in half along a diagonal, the two outside edges of each half are two
sides of the rhombus — so they must be equal.

Fold and check: Cut out one of these triangles, then fold it in half so the two equal
sides lie on top of each other. The two bottom corners land exactly on each other.
That shows the triangle also has 2 equal angles.

Q3 Trace the isosceles triangles on a paper and cut them out. Fold them in half. What
do you notice about their angles?

The two bottom angles land exactly on top of each other. So each isosceles triangle has 2
equal angles.
Steps

1. Trace the triangle on plain paper and cut it out.
2. Find the two equal sides with your ruler.
3. Fold so that one equal side lies exactly along the other. Press the fold flat.
4. Look at the two bottom corners. They cover each other completely — no bit sticks out.
5. Open the paper. The fold line runs from the top corner straight down to the middle of the
base.

Page 33 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
The two marked angles are equal e m.
m l as
m .co a g
l a se
a g

co m
ag
folded in half:
m .
as e the two corners
a g l
sit on each other

co m
em.
as
fold along the red line
m l
m .co a g
l a se
g
Folding is a way of measuring. If two angles cover each other, they are equal.
a
om a s
. c agl
Why folding proves it: When you fold, every point on one half lands on the
matching point of the other half. Ifm
a s e the two bottom corners land on each other, the

athel same size. This works even though we have not
two angles must be exactly g
measured a single degree.

co m
m .
e
omopposite the equal sides — the two at the base. aglas
Remember this pair: 2 equal sides → 2 equal angles. The equal angles are always
the.ctwo
m
ase
agl
se m
com
Is it possible to make a triangle where all three sides are equal (equilateral
g l a
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Q4

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ase
triangle)?

agl

co m
No — not with the pieces of this rhombus. Every triangle you build has exactly 2 equal sides,

m .
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never 3.
m l as
.co a g
Measure and see (use the printed rhombus; your cut-out will give the same story):

se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

com
m .
m ase
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a g l Page 34 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

TRIANGLE MADE THE TWO SLANTING THE BASE ALL THREE
SIDES EQUAL?

Tall one (base = short side of the rhombus short diagonal — a little No
diagonal) longer

Wide one (base = long side of the rhombus long diagonal — much No
diagonal) longer

Why it happens: The two slanting sides are always two sides of the rhombus, so
they are equal to each other. The base is always a diagonal, and in this rhombus
neither diagonal is the same length as a side. So you get two equal sides and one
odd one — an isosceles triangle, not an equilateral one.

That is why the book gives you a separate equilateral triangle on the cut-out
sheet at the end. You need one for the next activity, and the rhombus cannot give
you one.

Think about it: If a rhombus had its short diagonal exactly as long as its sides, then
half of it would be equilateral. Measure your cut-out carefully and see how close it
comes.

Q5 Is it possible to make a triangle where all three sides are unequal?

No. Every triangle you can build from these pieces has 2 equal sides, so you can never get three
sides that are all different.
Try all the ways of joining two pieces along a whole side:

1. Join along the long sides → you get the tall triangle. Two sides equal.
2. Join along the short sides → you get the wide triangle. Two sides equal.
3. Join along the slanting sides → you get a kite or a rectangle. Those are four-sided shapes,
not triangles at all.

Ways of getting a triangle: 2

Both of them: isosceles (2 equal sides)

Triangles with no equal sides: 0

Page 35 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Why it happens: Whenever two pieces join to make a triangle, one piece is the
mirror of the other. A mirror always makes matching pairs, and that is what gives the
two equal sides. To get a triangle with three different sides you would need pieces
that are not mirror images.

New word: A triangle with no two sides equal is called a scalene triangle. You will
make one in the next activity, by cutting an equilateral triangle in half.

Try This — Page 97
Angles of an equilateral triangle, and the scalene triangle

TRY THIS

Q1 Cut the equilateral triangle provided at the end of the book. Check if all the angles
of an equilateral triangle are equal — just like you did with the isosceles triangle.

Yes. All three angles of an equilateral triangle are equal.
How to check by folding

1. Cut out the equilateral triangle. Its three sides are all the same length — measure them to
be sure.
2. Fold once. Bring corner A onto corner B. Corner A sits exactly on corner B, so angle A =
angle B.
3. Open it out and fold again, this time bringing corner B onto corner C. They match too, so
angle B = angle C.
4. Put the two results together. If A = B and B = C, then all three are equal.

Page 36 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Two folds are enough to check all three
angles A

Fold 1 (red): A lands on B
Fold 2 (purple): B lands on C

So angle A = angle B = angle C

B C

Each fold line goes from a corner to the middle of the opposite side.

Angle A = Angle B (first fold)

Angle B = Angle C (second fold)

So Angle A = Angle B = Angle C

Each angle = 180 ÷ 3 = 60

Why it must be so: Equal sides give equal angles. An equilateral triangle has three
equal sides, so it has three equal angles. And since the three angles of any triangle
add up to 180, each one here is 60.

Now you know why six of them fit round a point: 60 × 6 = 360. That is the same
fact you found in the Find Out box.

Q2 Now, cut the equilateral triangle in half. How many sides of each new triangle are
equal?

None. All three sides of each new triangle are different lengths. A triangle like that is called
a scalene triangle.
What happens when you cut

Page 37 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

1. Fold the equilateral triangle in half and cut along the fold. You get two smaller triangles,
and they are alike.
2. Look at one of them. Its longest side is a whole side of the big triangle.
3. Its shortest side is half of another side of the big triangle — so it is only half as long.
4. Its third side is the cut line, which is neither of those two lengths.
5. Measure all three. You get three different numbers. No two sides are equal.

One equilateral triangle → two scalene
triangles

longest side
(the cut)

cut along the red line half a side

The half-triangle also has a square corner where the cut meets the base.

Why the sides are all different: The cut goes from a corner down to the middle of
the opposite side. So one new side is a full side of the big triangle, another is only
half of a side, and the cut itself is shorter than a full side but longer than half of one.
Three different lengths.

New word: Scalene means “uneven”. A scalene triangle has no equal sides.

Q3 Check in scalene triangles whether any two or more angles are equal?

No two angles of a scalene triangle are equal. All three angles are different.
How to check by folding

1. Take one of your half-triangles.

Page 38 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

2. Try to fold one corner onto another. Try every pair: first with second, second with third,
co m
first with third.
e m.
c o m g l as
.
3. Watch what happens. In every try the two corners do not cover each other — one always

e m past the other. a
s
pokes out
ano two angles are equal.
agl
4. So

co m
. ag
In the half of an equilateral triangle:
e m
the square corner = 90
g l as
the corner kept from the big triangle = 60 a

co m
m.
the third corner = 180 − 90 − 60 = 30

m as e
l
90, 60 and 30 are all different ✓
.co a g
a s em
a gl The rule behind it: Equal sides go with equal angles, and unequal sides go with
s
unequal angles. A scalene triangle has no two sides equal, so it can have no two
m a
.co agl
angles equal either.

se m
g l a
a
The three families together: equilateral — 3 equal sides, 3 equal angles. Isosceles

m
— 2 equal sides, 2 equal angles. Scalene — no equal sides, no equal angles. The
sides and the angles always tell the same story.
. co
e m
m l as
.co a g
a s em
a l
gShapes from the Rhombus — Page 98
The kite, the parallelogram and the rectangle
se m
com g l a
m . a
ase
agl
Q1 How many different 4-sided shapes (quadrilaterals) can you make? Here are three
possible shapes.

co m
m .
as e
com l
You can make 4 different four-sided shapes: a kite, a rectangle, and two different
. a g
se m
parallelograms. The book shows three of them.

a
agl
Remember, each piece is the same right-angled triangle. Call its sides the short side, the long
side and the slanting side. Two pieces can be joined along any pair of matching sides, and the
.c
s e m
m a
second piece can be placed either as a mirror or turned half a turn.

e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 39 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

JOIN ALONG SECOND PIECE PLACED AS SHAPE YOU GET

slanting side a mirror Kite

slanting side turned half a turn Rectangle

short side turned half a turn Parallelogram (long one)

long side turned half a turn Parallelogram (flatter one)

kite rectangle parallelogram parallelogram

Four quadrilaterals from the same two pieces. The dotted line is the join.

Why the shape changes: Only two things can change — which side you glue, and
whether you flip the piece or turn it. A flip makes matching pairs next to each other,
and that gives a kite. A half turn sends each side to the opposite side, and that gives
a parallelogram or a rectangle.

Check it yourself: Whatever you make, its area is the same — two pieces are two
pieces. Only the outline changes.

Page 40 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q2 Have you made a shape like the one shown on the right? This shape is called a kite.
What do you notice about the sides of a kite? Side 1 = Side 2. Similarly, Side 3 = Side
4. These sides are called adjacent sides.

1 2

4 3

The shape shown on the right on page 98, with its four sides numbered.

A kite has two pairs of equal sides, and the equal sides are next to each other. Sides that
are next to each other are called adjacent sides.

Page 41 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Side 1 = Side 2 (the two short
ones, side by side)

1 2

4 3

Side 3 = Side 4 (the two long ones,
side by side)
In a kite the equal sides touch each other. In a parallelogram the equal sides are opposite each other.

Side 1 = Side 2 (a pair of adjacent sides)

Side 3 = Side 4 (another pair of adjacent sides)

But Side 1 is not equal to Side 3.

Why the kite comes out this way: You made it by folding one piece over onto the
other like a mirror. The mirror line is the dotted line down the middle. Whatever is on
the left is copied on the right — so the two short sides match and the two long sides
match.

Page 42 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Where you have seen it: A real paper kite you fly at Makar Sankranti is exactly this
shape. Two short sides at the top, two long sides at the bottom, and the stick runs
down the mirror line.

Q3 Measure the sides of each of these two quadrilaterals A and B. What do you notice?
Are there any pairs of sides that are equal? Which pairs are equal — adjacent or
opposite?

In both A and B, the sides that are equal are the opposite ones, not the adjacent ones.
What your ruler will show

1. Measure the four sides of A. The top and the bottom come out the same. The left and the
right come out the same.
2. Measure the four sides of B. Same story — top matches bottom, left matches right.
3. Compare a side with the one next to it. They are not equal.
4. So the equal pairs are the opposite pairs.

equal

equal
A — a parallelogram B — a rectangle

Both shapes have their opposite sides equal. B has square corners as well.

In A: top = bottom, left = right → opposite sides equal

In B: top = bottom, left = right → opposite sides equal

A quadrilateral whose opposite sides are equal is called a parallelogram.

Page 43 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

co m
m.
Why they are not kites: You built A and B by turning the second piece half a turn

as e
om end up opposite each other, not next to each other. l
instead of flipping it. A half turn sends each side across to the far side, so the
matching.c a g
m
sides
as e
a g l
Kite or parallelogram? Ask one question: Are the equal sides touching (adjacent)

co m
ag
or facing (opposite)? Touching → kite. Facing → parallelogram.
m .
ase
a g l
What types of angles do quadrilaterals A and B have? Which angles are equal in
m
Q4

co
m.
each of the above parallelograms?

o m l a se
ANSWER c
. a g
m
e opposite angles are equal. In B all four angles are equal, and each one is a right
InaAsthe
l
agangle — so B is a rectangle.
m a s
.co agl
SHAPE TYPES OF ANGLES WHICH ANGLES ARE EQUAL NAME

a s emThe two sharp ones are equal; the two blunt
A
a
Two sharp (acute) angles and
two blunt (obtuse) angles
gl ones are equal — and each pair is opposite
Parallelogram

com Rectangle
.
B Four right angles All four are equal

e m
m l as
.co
How to check without a protractor
a g
a s em
a g l 1. Take the corner of a fresh sheet of paper. That corner is a right angle.
2. Hold it into each corner of B. It fits exactly every time — so all four angles of B are right
angles.
se m
o m l a
.c ag
3. Hold it into the corners of A. Two corners are smaller than the paper corner and two are
larger. So A has no right angle. em
g l as
4. Trace one sharp corner of
a A on tracing paper and slide it across to the opposite corner. It
fits — those two angles are equal.

c o m
.
m has opposite sides
Why a rectangle is a special parallelogram: A rectangle already
s e
. om which is what makes something a parallelogram.
cequal, a gItlahas one extra gift on top
a s em — all its angles are right angles. So every rectangle is a parallelogram, but not every
agl parallelogram is a rectangle.
c
m .
m a s e
m . co
The family tree so far: quadrilateral → parallelogram (opposite sides equal) →
e agl
l as
rectangle (and all angles right angles) → square (and all sides equal too).
g
a

co m
m .
m ase
.co


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Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Shapes from the Rhombus — Page 99
Drawing on a grid, and building with 3 and 4 pieces

Q1 In the grid given below, draw two different kites and parallelograms each.

The grid given in the book, page 99.

Use the corners of the grid squares as your points. Here are two kites and two parallelograms
you can copy.
How to draw a kite on the grid

1. Draw a straight line down the grid. This is the mirror line.
2. Mark a point on the line near the top and another near the bottom.
3. Mark two more points, one to the left and one to the right of the line, both the same
number of squares away and on the same grid row.
4. Join the four points in order. The two upper sides are equal and the two lower sides are
equal — that is a kite.

How to draw a parallelogram on the grid

1. Draw the bottom side — say 5 squares long.
2. Step across and up from one end, for example 2 squares right and 3 squares up. Mark the
point.
3. Make exactly the same step from the other end of the bottom side. Mark that point too.
4. Join the four points. The top and bottom are equal, the two slanting sides are equal — that
is a parallelogram.

Page 45 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

kite 2 (a rhombus)
kite 1

parallelogram 2 parallelogram 1

Two kites (blue and green) and two parallelograms (yellow and pink), all drawn on grid corners.

Why the grid makes it easy: Counting squares is a way of measuring without a
ruler. If you count the same number of squares on both sides of the mirror line, the
two sides must be equal.

Did you notice? Kite 2 has all four sides equal. A kite with all four sides equal is a
rhombus. So a rhombus is a special kind of kite.

Page 46 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q2 Now, use 3 triangles from the rhombus to form shapes. How many sides do each
one of them have? Using 3 triangular pieces of the rhombus, try creating a (a) 3-
sided shape, (b) 4-sided shape, and (c) 5-sided shape.

The rhombus and its four triangular pieces.

(b) and (c) work easily. (a) does not — three of these pieces will not close up into a triangle.
Here is what happens when you actually try.

Three pieces, two different outlines

the notch here
makes the 5th side

(b) 4-sided shape (c) 5-sided shape

Two pieces make a rectangle; where the third piece is stuck on decides how many sides the whole
shape has.

(b) A 4-sided shape

1. Put two pieces together along their slanting sides to make a rectangle.
2. Take the third piece and place its long side exactly along the top of the rectangle.
3. The outline now has 4 corners — it is a four-sided shape (a trapezium).

Page 47 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

(c) A 5-sided shape

1. Make the same rectangle from two pieces.
2. Place the third piece on the top edge, but this time put its short side along the top, starting
from the left corner.
3. Because the short side is shorter than the top of the rectangle, a step (a notch) is left. That
step adds an extra corner, so the outline has 5 sides.

(a) A 3-sided shape

1. Two pieces alone already make a triangle — that is half the rhombus.
2. Now try to add a third piece anywhere along its edges.
3. Every time, one more corner appears, so you end up with 4 or 5 sides, never 3.
4. The reason: for the outline to stay a triangle the new piece must have a side exactly as long
as the edge it is covering, and none of its sides is the right length.

Worth knowing: If your rhombus were made of two equilateral triangles (a wider,
flatter rhombus), then three pieces would close up into a triangle. Ask your teacher
for such a rhombus and try it — the answer to a shape question can depend on the
exact shape you start with.

Q3 Which of these shapes can be made with all 4 pieces? Try and find out. (a) Square (b)
Rectangle (c) Triangle (d) Pentagon (5-sided) (e) Hexagon (6-sided) (f) Octagon (8-
sided)

The rhombus and its four triangular pieces.

(b) Rectangle, (c) Triangle, (d) Pentagon and (e) Hexagon can be made. (a) Square and (f)
Octagon cannot.

Page 48 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
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co m
m.
SHAPE CAN YOU REASON

m
MAKE IT?
as e
.cNoo a g l
(a) Square
a s em Two pieces make a rectangle whose sides are the two different half-lines

ag l of the rhombus. Putting two such rectangles side by side still gives a
rectangle, never a square.

co m
. ag
(b) Yes Two pieces make one rectangle; place a second such rectangle beside it,
Rectangle
se m
long side to long side.

l a
(c) Yes ag make one big triangle with every side exactly twice as long as
Four pieces
Triangle the piece's sides.

co m
em.
(d) Yes Build the rectangle from three pieces and stick the fourth on so that it

m l as
.co
leaves one step.
g
Pentagon

em a
a s
glHexagon
(e) Yes Stick the pieces on so that two steps are left in the outline.

a
om a s
agl
(f) No Four triangles have 4 × 3 = 12 edges in all. Joining them uses up at least 3

. c
m you can get.
Octagon pairs, that is 6 edges. So at most 6 edges are left on the outside — six

a s e
sides is the most

agl
Two of the shapes you can definitely build
co m
m .
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se m
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a
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m . a
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a g l Page 49 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Did you know? If a rhombus happened to have one line exactly twice as long as the
other, then its four pieces would make a square. Measure your rhombus and see how
far it is from that.

Tangram — Page 99
Naming the seven pieces and building shapes with them

TANGRAM

Q1 Look at the tangram set given at the end of your textbook. Cut out all the shapes.
Name them.

The tangram set given at the end of the textbook.

A tangram is one square cut into 7 pieces. Here they are, with their names.

Page 50 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

HOW MANY NAME OF THE PIECE NUMBER OF SIDES

2 Large triangle 3

1 Medium triangle 3

2 Small triangle 3

1 Square 4

1 Parallelogram 4

7 in all 5 triangles + 2 four-sided pieces

The seven tangram pieces (not drawn to exact size)

2 large triangles 1 medium triangle
2 small triangles

1 square 1 parallelogram

Five of the seven pieces are triangles. The other two are a square and a parallelogram.

Handle with care: Cut along the printed lines slowly, and keep all 7 pieces in one
envelope. If even one piece is lost, most of the puzzles cannot be made.

Q2 How are they same or different from each other?

Same: five of the seven pieces are triangles, and all five are exactly the same shape — a right-
angled triangle with two equal sides. Different: they come in three sizes, and two pieces are not
triangles at all.

Page 51 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

WHAT IS THE SAME WHAT IS DIFFERENT

All 5 triangles have one square corner They come in 3 sizes — large, medium, small

All 5 triangles have 2 equal sides The square and the parallelogram have 4 sides, not 3

Every corner in the whole set is 45, 90 The square has 4 equal sides; the parallelogram does not
or 135

The 2 large pieces match each other; so The medium triangle, the square and the parallelogram cover the same
do the 2 small ones amount of space, but look different

2 large triangles are equal to each other

2 small triangles are equal to each other

1 medium triangle = 1 square = 1 parallelogram = 2 small triangles (same amount of space)

1 large triangle = 4 small triangles

Why this matters for the puzzles: Because the medium triangle, the square and
the parallelogram all cover the same space, you can often swap one for another.
That is why the same picture can be made in more than one way.

Check it yourself: Place the two small triangles on top of the square piece, and then
on top of the medium triangle. Each time they cover it exactly.

Page 52 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Q3 What do you notice about the angles of each of the shapes?

The tangram set given at the end of the textbook.

Every corner in the whole tangram set is one of only three sizes: 45, 90 or 135.

PIECE ITS ANGLES TOTAL

Every triangle (all 5) 90, 45, 45 180

Square 90, 90, 90, 90 360

Parallelogram 45, 135, 45, 135 360

How to see it without a protractor

1. Take the corner of a sheet of paper — that is a right angle, 90.
2. Fit it into the biggest corner of any triangle. It matches exactly.
3. Now fold that paper corner in half. The new corner is 45.
4. Fit it into the two sharp corners. Both match, so both are 45.
5. For the parallelogram's blunt corner, hold the paper right angle plus the folded 45
together — 90 + 45 = 135. It matches.

Why only these three sizes: The whole tangram is cut from one square. Every cut is
either along a side or along a diagonal, and a diagonal cuts a right angle exactly in
half. So every angle in the set is 90, or half of it (45), or one and a half of it (135).

Page 53 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

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a g l Page 54 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Why the pieces always fit: Because so many sides are exactly the same length, a
piece can be laid flat against another with no gap. If the lengths did not match, the
tangram would not work at all.

Check it yourself: Line up the slanting side of a small triangle against the short side
of the medium triangle. They end at the same place.

Q5 Now, use some or all of the pieces of your tangram set to make the following
shapes. There may be more than one way to do it.

The four shapes to be made, page 99.

The four outlines in the book are a parallelogram, a rectangle, a triangle and a trapezium.
Here is one way to make each.

Page 55 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Rectangle — the square piece + 2 small triangles
Triangle — the 2 large triangles, joined along a short side

Parallelogram — the 2 large triangles,
Trapezium
one turned
— thehalf
square
a piece + 2 small triangles
turn
The thick outline is the shape you are asked to make.
The thin lines show where the pieces meet inside it.

Four target shapes, four simple solutions. Other solutions exist — try to find them.

Steps for each one

1. Triangle: Take the two large triangles. Put a short side of one exactly against a short side of
the other, like a mirror. You get a bigger triangle.
2. Parallelogram: Take the same two large triangles, but turn one of them half a turn before
joining. Now you get a parallelogram.
3. Rectangle: Join the two small triangles along their long slanting sides — they make a small
square. Put that beside the square piece.
4. Trapezium: Put the square piece in the middle. Stand one small triangle against its left side
and one against its right side, both sloping outwards.

Try more: Every one of these can also be made using all 7 pieces. That is much
harder — a good challenge for a rainy afternoon.

Which Shape Am I? — Page 100

Page 56 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Matching statements with the square, rectangle, rhombus and parallelogram

WHICH SHAPE AM I?

Q1 Match the statement with the appropriate shape: 1. All my angles are right angles,
but all my sides are not equal.

(Square) (Rectangle) (Rhombus)

(Parallelogram)

The shapes to choose from, page 100.

Rectangle.

The four shapes you are matching against

Square Rectangle Rhombus Parallelogram

Keep this row in front of you while you answer all seven statements.

All angles right angles → square or rectangle

But all sides are not equal → this rules out the square

So the answer is the rectangle

Page 57 of 90

Class 5 Maths Chapter 7 Shapes and Patterns AglaSem · NCERT Solutions

Why not the square: A square also has four right angles, but a square has all four
sides equal. The statement clearly says the sides are not all equal, so the square is
out.

Q2 2. All my sides are equal, but all my angles are not.

Rhombus.

All sides equal → square or rhombus

But all angles are not equal → this rules out the square

So the answer is the rhombus

Why it fits the rhombus: A rhombus is a “pushed-over square”. Its four sides stay
equal, but two of its corners become sharp and two become blunt. So the sides
match and the angles do not.

Picture it: Make a square with four sticks joined loosely at the corners, then push
one corner sideways. The sides do not change length, but the corners change size.
You now have a rhombus.

Q3 3. My opposite angles are equal, but my sides do not make a right angle.

Two shapes fit: the rhombus and the parallelogram.

Opposite angles equal → square, rectangle, rhombus or parallelogram

No right angles → this rules out the square and the rectangle

What is left: rhombus and parallelogram

Page 58 of 90

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Class 5 Maths Chapter 7 Shapes and Patterns
a g l AglaSem · NCERT Solutions

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Answer: rhombus and parallelogram

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a g l Page 59 of 90