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Math-Magic Solutions for Class-5, Chapter-3: How Many Squares?

The chapter How Many Squares? focuses on the areas of geometric shapes by measuring them with square grids and estimating how many squares a shape wpuld cover. It helps the students to understand the the concept of area through the idea of a square as a unit.

It covers the topics:

Areas of shapes measured in squares |
Estimation of the area of basic shapes |

Areas of shapes measured in squares |

Estimation of the area of basic shapes |

NCERT Math-Magic questions are answered in a simple and engaging manner. We have also related 'Learning Concepts and interactive worksheets with the solutions. Our 'Learning Beyond' segment caters to all the probable questions that a child might think out of curiosity.

Download Chapter 3 How Many Squares? in PDF format for free here.

**Chapter 3 How Many Squares?**

**Question 1 : **

a) Measure the side of the red square on the dotted sheet.

b) Draw here as many rectangles as possible using 12 such squares.

c) How many rectangles could you make?

d) Which of these rectangles has the longest perimeter?

e) Which of these rectangles has the smallest perimeter?

**Answer : **

a) Measure the side of the shaded square using a ruler. The side of the
red square is 1 cm.

b) All possible rectangles using 12 squares are shown below.

c) Count the number of rectangles drawn in the part b), there are 7
rectangles in all.

d) Observe the rectangles drawn in part b). Find the perimeter of each
rectangle by counting the number of sides of the small squares along
their boundary.

The perimeter of rectangle 1 = 26 cm

The perimeter of rectangle 2 = 26 cm

The perimeter of rectangle 3 = 14 cm

The perimeter of rectangle 4 = 14 cm

The perimeter of rectangle 5 = 14 cm

The perimeter of rectangle 6 = 14 cm

The perimeter of rectangle 7 = 16 cm

Thus, rectangles 1 and 2 have the longest perimeters.

e) Observe the perimeters of the rectangles in part d). Rectangles 3, 4,
5, and 6 have the smallest perimeters.

**Question 2 : **

a) How many squares of one centimetre side does stamp A
cover?.............

And stamp B cover?.....................

b) Which stamp has the biggest area?

How many squares of side 1 cm does this stamp cover?

How much is the area of the biggest stamp?

c) Which two stamps have the same area?

How much is the area of each of these stamps?

d) The area of the smallest stamp is .............square cm.

The difference between the area of the smallest and the biggest stamp
is.............square cm.

**Answer : **

a) Observe the given picture and count the number of small squares
covered by the stamp A, and the stamp B.

Stamp A covers 18 squares.

Stamp B covers 8 squares.

b) Observe the given picture, the area of a stamp is the number of
small squares covered by that stamp. Therefore,

area of stamp A =18 square cm

area of stamp B = 8 square cm

area of stamp C = 6 square cm

area of stamp D = 12 square cm

area of stamp E = 4 square cm

area of stamp F = 12 square cm

Hence, stamp A has the biggest area. It covers 18 squares of side 1
cm. The area of stamp A is 18 square cm.

c) Observe the areas of stamps in part b). Stamps 'D' and 'F' have the
same area, 12 square cm each.

d) From the part b), the area of the smallest stamp is 4 square cm.

This stamp covers 4 squares of side 1 cm.

Since the area of the biggest stamp is 18 square cm, and the area of
the smallest stamp is 4 square cm, subtract 4 from 18 to get the
difference.

18 – 4 = 14

Therefore, the difference between the area of the smallest and the
biggest stamp is 14 squares cm.

**Question 3 : **

a) Which has the bigger area — one of your footprints or the page of
this book?

b) Which has the smaller area—two five-rupee notes together or a
hundred rupee note?

c) Look at a 10 rupee-note. Is its area more than hundred square cm?

d) Is the area of the blue shape more than the area of the yellow
shape? Why?

e) Is the perimeter of the yellow shape more than the perimeter of the
blue shape? Why?

**Answer : **

a) Do it by yourself.

b) Do it by yourself.

c) Do it by yourself. Put a hundred rupee note on a square grid paper
and count the number of squares covered by the note.

d) Observe the given images, the yellow square is divided into two
triangles, these triangles are rearranged and made bigger blue triangle.
Therefore, the area of the blue shape and the area of the yellow shape
are equal.

e) Observe the given images. The triangle’s base is double in length
than the length of a side of the square. The other two sides of the
triangle are larger than the side of the square. So, the perimeter of the
triangle will be more than the sum of the four sides of the square.
Therefore, the blue shape has larger perimeter.

**Question 4 : **Write the area (in square cm) of the shapes below.

**Answer : **To find the area of the given shapes, count the squares covered by
them
in the following manner.

• If a square is fully covered, count is as 1.

• If a square is more than half covered, count it as 1.

• If a square is exactly half covered, count it as 1/2.

• If a square is less than half covered, count it as 0.

**Shape A:**

3 full squares ⇒ 3

3 more than half squares ⇒ 3

3 less than half squares ⇒ 0

Area of shape A = 3 + 3 + 0 = 6 square cm.

**Shape B:**

4 full squares ⇒ 4

8 half squares ⇒ 4

4 less than half squares ⇒ 0

Area of shape B = 4 + 4 = 8 square cm.

**Shape C:**

2 full squares ⇒ 2

2 more than half squares ⇒ 2

2 less than half squares ⇒ 0

Area of shape C = 2 + 2 + 0 = 4 squares cm.

**Shape D:**

5 full squares ⇒ 5

2 half squares ⇒1

Area of shape D = 5 + 1 = 6 squares cm.

**Shape E:**

18 full squares ⇒ 18

6 half squares ⇒ 3

Area of shape E = 18 + 3 = 21 square cm.

**Shape F:**

4 full squares ⇒ 4

4 more than half squares ⇒ 4

4 less than half squares ⇒ 0

Area of shape F = 4 + 4 + 0 = 8 square cm.

**Question 5 : **The blue triangle is half of the big rectangle. Area of the big
rectangle
is 20 square cm. So the area of the blue triangle is...........square cm.

**Answer : **

Area of big rectangle = 20 square cm

Since the blue triangle is half of the big rectangle,

area of the blue triangle = (20 ÷ 2) = 10 square cm.

**Question 6 : **Ah, in it there are two halves of two
different rectangles!

Now you find the area of the two rectangles Sadiq is talking about.
What is the area of the red triangle? Explain.

**Answer : **Observe the given picture, the orange rectangle has covered 12
squares. Therefore,

area of the orange rectangle = 12 square cm.

The green rectangle has covered 8 squares. Therefore,

Area of the green rectangle = 8 square cm.

Since the triangle has covered half of the orange rectangle and half of
the green rectangle, the area of the triangle is

1 /2 (area of the orange rectangle area of thegreen rectangle)

= 1/2 (12+8)=20/2 = 10

So, the area of the triangle is 10 square cm.

**Question 7 : **Suruchi drew two sides of a shape. She asked Asif to complete the
shape with two more sides, so that its area is 10 square cm.

He completed the shape like this.

Is he correct? Discuss

Explain how the green area is 4 square cm and the yellow area is 6
square cm.

**Answer : **
Following are the rules to find area by counting the squares.

• If a square is fully covered, count is as 1.

• If a square is more than half covered, count it as 1.

• If a square is exactly half covered, count it as 1/2.

• If a square is less than half covered, count it as 0.

The green area has:

2 full squares ⇒ 2

4 half squares ⇒ 2

Therefore, the green area is 2 + 2 = 4 square cm.

The yellow area has:

3 full squares ⇒ 3

3 more than half squares ⇒ 3

3 less than half squares ⇒ 0

Therefore, the yellow area is 3 + 3 + 0 = 6 square cm.

The total area is 4 + 6 = 10 square cm. Hence, Asif is correct.

**Practice Time**

**Question 8 : **This is one of the sides of a shape. Complete the shape so that
its area
is 4 square cm.

**Answer : **Following is the triangle, covered 2 full squares and 4 half squares.
So,
its area is 2 + 2 = 4 square cm.

**Question 9 : **Two sides of a shape are drawn here. Complete the shape by drawing
two more sides so that its area is less than 2 square cm.

**Answer : **The correct answer is:

**Question 10 : ** Here is a rectangle of area 20 square cm.

a) Draw one straight line in this rectangle to divide it into two equal
triangles. What is the area of each of the triangles?

b) Draw one straight line in this rectangle to divide it into two equal
rectangles. What is the area of each of the smaller rectangles?

c) Draw two straight lines in this rectangle to divide it into one
rectangle and two equal triangles.

• What is the area of the rectangle?

• What is the area of each of the triangles?

**Answer : **

a) Join opposite corners of the rectangle by drawing a straight line.

Since the area of the rectangle is 20 square cm, the area of the tringle
is

20/2 = 10 ㎠

b) You can draw a straight line along the length of the rectangle or
along its breadth.

Since the area of the big rectangle is 20 square cm, the area of the
smaller rectangle is

20/2 = 10 ㎠

d) Draw the two straight lines as shown below.

Since the smaller rectangle is half of the big rectangle, the area of the
smaller rectangle is

20/2 = 10 ㎠
Since the triangle is half of the smaller rectangle, the area of the
triangle is 10/2 = 5 ㎠.

**Question 11: **Puzzle with five squares.

a) How many different shapes can you draw?

b) Which shape has the longest perimeter? How much?

c) Which shape has the shortest perimeter? How much?

d) What is the area of the shapes? _______ square cm. That’s simple!

**Answer : **

a) Do it by yourself. You can draw so many shapes using five
squares. A sample answer is given below.

b) All the shapes have longest perimeter except the shape 4, the
longest perimeter is:

1+1+1+1+1+1+1+1+1+1+1+1 = 12 cm

c) The shape 4 has the smallest perimeter. The smallest perimeter is:

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1= 10 cm

d) There are 12 shapes, each with five complete squares. Therefore,
the area of each shape is 5 square cm.

**Question 12: **
Ziri tried to make some other tiles. She started with a square of 2 cm
side and made shapes like these.

Look at these carefully and find out:

Which of these shapes will tile a floor (without any gaps)? Discuss.

**Answer : **
Tile is a pattern where there is no gap left. Observe the given shapes
carefully and find out which will tile a floor without any gap. The
correct answer is the shape C and shape D. Following are the designs..