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

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.

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.

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.

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.

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.

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.

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.

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

The correct answer is:

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.

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..