NCERT Solutions for Class 5 Maths Chapter 3 How Many Squares?

1. 

Measure the side of the red square on the dotted sheet. Draw here as many rectangles as possible using 12 such squares.

(i) How many rectangles could you make? ________

Ans: On the dotted sheet, the side of the square is 1 cm.

We can make 7 possible rectangles in the following figure.

We can make 7 rectangles.

• 2 rectangles are of size 1 $\times$ 12 centimetre.

• 1 rectangle is of size 2 $\times$ 6 centimetre.

• 4 rectangles are of size 3 $\times$ 4 centimetre

Hence, number of rectangles = 2 + 1 + 4 = 7

2. Each rectangle is made out of 12 equal squares, so all have the same area, but the length of the boundary will be different.

(i) Which of these rectangles has the longest perimeter?

Ans:  To find the perimeter of the rectangle for 1 x 12 cm we have to do to following steps:

By using Formula : 2(length + breadth)

= 2(1 + 12)

= 2 $\times$ 13 = 26 cm

To find the perimeter of a rectangle for 2 $\times$ 6 cm we have to do to following steps:

By using Formula : 2(length + breadth)

= 2(2 + 6)

= 2 $\times$ 8 = 16 cm

To find the perimeter of the rectangle for 3 $\times$ 4 cm we have to do to following steps:

By using Formula : 2(length + breadth)

= 2(3 + 4)

= 2 $\times$ 7 = 14 cm

(ii) Which of these rectangles has the smallest perimeter?

Ans: The rectangle 3 $\times$ 4 cm has the smallest perimeter.

3. Measure Stamps
Stamp D covers 12 squares. Each square is of side 1 cm. So the area of stamp D is 12 square cm.

Look at these interesting stamps.

(a) How many squares of one centimetre side does stamp A cover and stamp B? ________

Ans: By using the formula:

Area = length $\times$ breadth

= 4 $\times$ 2 = 8 square cm

(b) Which stamp has the biggest area?

Ans: The biggest area is Stamp A.

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

Ans: This stamp covers 18 squares with sides of 1 cm.

(ii) How much is the area of the biggest stamp? _____ square cm.

Ans: 18 square cm is the area of the biggest stamp.

(c) Which two stamps have the same area? _____ 

Ans: Stamp ‘D’ and Stamp ‘F’ have the same area.

(i) How much is the area of each of these stamps? ____ square cm.

Ans: The area of Stamp ‘D’ is 12 square cm,

The area of Stamp ‘F’ is 12 square cm.

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

Ans: The area of the smallest stamp that is Stamp ‘E’ is length $\times$ breadth

= 2 $\times$ 2

= 4 $cm^{2}$

(i) The difference between the area of the smallest and the biggest stamp is _____ square cm.

Ans: To find the difference between the area of the smallest and the biggest stamp of square cm
Area of the Biggest Stamp - Area of the Smallest Stamp

= 18 - 4 = 14 square cm.

4. Guess

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

Ans: When the area of the footprints is compared to the area of the page of the textbook, the page of the textbook has a larger area than the footprints.

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

Ans: A hundred rupee note takes the smaller area by comparing the 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?

Ans: No, a 10 rupee note has an area size of less than 100 square centimetres.

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

Ans: No, the blue shape's area and the yellow shape's area are equal. By cutting out these shapes and using the square grid paper to calculate their areas, it is simple to figure out this.

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

Ans: No, the yellow shape's perimeter is smaller than the blue shape's. You can measure their limits with a ruler or a thread to prove this.

6. How Big is My Hand? 

Trace your hand on the squared sheet on the next page.

How will you decide whose hand is bigger - your hand or your friend’s hand?

What is the area of your hand? _______ square cm.

What is the area of your friend’s hand? _______ square cm.

Ans: Trace both of your hands and your friend's hands on a square piece of paper. Find how many of each type of square there are: full, half-filled, more than half-filled, and less than half-filled. Remove the squares that are less than half full. To find the area, add the number of squares that are filled and those that are more than half full.

Considering Figure A:

6 complete squares are present.

There are 7 half-filled squares.

6 squares are more than half-filled.

Area = 6 + (7/2) + 6 

= 6 + 3.5 + 6 

= 15.5 square cm is the area.

Considering Figure B:

13 complete squares are present.

There are 4 half-filled squares.

13 squares are more than half-filled.

Area = 13 + (4/2) + 13 

= 13 + 2 + 13 

 = 28 square cm is the area.

7. My Footprints

My footprint is longer!

But my footprint is wider. So whose foot is bigger?

(i) Whose footprint is larger yours or your friend’s?

Ans: My footprint is larger.

(ii) How will you decide? Discuss.

Ans: We can decide by tracing both footprints on a squared sheet of paper and counting the number of complete squares, half-filled squares, and more than half-filled squares inside each footprint. The one with the larger total area is the bigger footprint.

(iii) Is the area of both your footprints the same?

Ans: No, the area of both footprints is not the same.

8. Guess which animal’s footprint will have the same area as yours. Discuss.

Here are some footprints of animals — in actual sizes. Guess the area of their footprints.

Ans:  A monkey's footprint may be similar in size to mine.

Hen's Footprint:

Number of complete squares: 5

Number of half-filled squares: 4

Number of more-than-half-filled squares: 3

Area of hen's footprint:

= 5 + (4/2) + 3 

= 5 + 2 + 3 

= 10 square units

Dog's Footprint:

Number of complete squares: 8

Number of half-filled squares: 6

Number of more-than-half-filled squares: 5

Area of dog's footprint; 

= 8 + (6/2) + 5 

= 8 + 3 + 5 

= 16 square units

From this example, you can see the dog's footprint has a larger area than the hen's footprint.

9. Make big squares and rectangles like this to find the area faster.

Ans: 

Area of P: 

= 5 + 2/2 + 7

= 5 + 1 + 7 

= 13 square cm

Area OF Q:

= 7 + 2/2 + 4

= 7 + 1 + 4 

= 12 square cm

Area of R:

= 6 + 0/2 + 9

= 6 + 9 

= 15 square cm

Area of S:

= 4 + 0/2 + 6

= 4 + 6 

= 10 square cm

Area of T:

= 96 + 4/2 + 21

= 96 + 2 + 21 

= 119 square cm

10. How Many Squares in Me?

(i) What is the area of this triangle?

Ans: The area of this triangle is 1 square centimetre.

(ii) The triangle is half the rectangle of area 2 square cm. So its area is ___ square cm.

Ans: The triangle is half the rectangle of area 2 square cm. So its area is 1 square cm.

(iii) Is this shape half of the big rectangle?

Ans: Yes, the above shape is half of the big rectangle.

(iv) Hmmm…… So its area is _____ square cm.

Ans: So its area is 4 square cm.

10.  Write the area (in square cm) of the shapes below.

Ans:

11. Sameena: Both the big triangles in this rectangle have the same area.

Sadiq: But these look very different.

(i) The blue triangle is half of the big rectangle. The area of the big rectangle is 20 square cm. So the area of the blue triangle is _______ square cm.

Ans: According to the question, The area of the triangle blue is $\dfrac{1}{2}\times 20$ = 10 square cm.

(ii) And what about the red triangle?

Ans: According to the question, The area of the triangle red is half of the area of a rectangle.

So, the Area of the triangle red is 10 square cm and there are two different rectangles with two halves.

(iii) Now find the area of the two rectangles in the figure. What is the area of the red triangle? Explain.

Ans: From the figure which is given, we can say that,

Given, 

The orange rectangle = 12 squares

Then, the green rectangle = 8 squares

So, the area of the orange rectangle is 12 square cm

So, the area of the green rectangle is 8 square cm 

Now, the area of the orange portion of the triangle = 12/2 = 6 square cm

The area of the yellow portion of the triangle 

= 8/2 

= 4 cm square

Therefore, the area of the red triangle 

= 6 + 4

= 10 square cm.

12. Complete the Shape 

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

(i) How did you do this?

Ans: Well, that's simple! The green area, which is 4 square centimetres, is seen. The yellow, 6 square centimetre area is below it. My form so it's 10 square centimetres in area.

(ii) Is he correct? Discuss.

Ans: Yes, he is correct about the above solution.

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

Ans: To find the area of the green portion we have to solve by using,

Area of green portion = Number of complete squares + Number of half-filled squares

=$2+\dfrac{1}{2}\times 4$ 

= 2 + 2 

= 4 square cm

To find the Area of the yellow portion = Number of complete squares + Number of half-filled squares + Number of more than half-filled squares

= $3+\dfrac{1}{2}\times 2+2$

= 3 + 1 + 2 

= 6 square cm

13. Suruchi : Oh, I thought of doing it differently! If you draw like this, the area is still 10 square cm.

(i) Is Suruchi correct? How much is the blue area? Explain.

Ans: Suruchi is right, 

The area of the green triangle = Half of area of rectangle measuring $4\times 2 = \dfrac{1}{2}\times 4\times 2 = 2\times 2$

=4 cm square

Area of the blue triangle = Half of the area of the rectangle measuring $ 4\times 3$

=  $\dfrac{1}{2}\times 4\times 3$

= $2\times 3$

= 6 square cm.

thereby, 4 + 6 = 10 square centimetres is the total area.

(ii) Can you think of some other ways of completing the shape?

Ans: Thus, a rectangle with a 12 square centimetre area can be created. We can then draw as many triangles in it as we can after that. It can be seen in the figure next to it.

Practice time

14. This is one of the sides of a shape. Complete the shape so that its area is 4 square cm.

Ans:

Finding the completed shape = 2 complete square + 4 half square

=$2+\left ( \dfrac{1}{2}\times 4 \right )$

=2+2

=4 $cm^{2}$

15. 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.

Ans: 

16. Here is a rectangle of area 20 square cm.

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

Ans: 

Area of rectangle:

= $10\times 2$

= 20 square cm

Then, the area of two equal triangles:

= $\dfrac{20}{2}$

= 10 square cm

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

Ans: 

Area of big rectangle = 10 square cm

The area of each of the smaller rectangles is $\dfrac{20}{2}$

= 10 square cm

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

(a) What is the area of the rectangle?

Ans: Area of Rectangle = (Length $\times$ Breadth)

= 2 $\times$ 5

= 10 square cm

(b) What is the area of each of the triangles?

Ans:  Area of each Triangle = $\frac{1}{2}$ $\times$ Area of the smaller Rectangle

= $\frac{1}{2}$ $\times$ 10

= 5 square cm

Puzzles with Five Squares

17. Measure the side of a small square on the squared paper. Make as many shapes as possible using 5 such squares. Three Eire drawn for you.

a) How many different shapes can you draw? ___________

Ans: I can draw 12 shapes, by using 5 squares.

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

Ans: The smallest perimeter out of 12 shapes is Shape 4, it has 12 square cm.

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

Ans: Shape 4 has the smallest perimeter of all 12 shapes, and Shape 4 has 10 cm square.

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

Ans: The total area of the shapes is 5 square cm.

Practice time

18 . 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.

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

Ans: Shapes C and D may create a floor without any gaps by following the provided figure.

Conclusion

NCERT Class 5 Maths Chapter 3 Solutions of "How Many Squares" offer a comprehensive and insightful exploration of squares and their arrangements. These solutions provide step-by-step explanations and examples to help students count the number of squares in various patterns and grids. By mastering the concept of squares, students enhance their problem-solving skills and geometric understanding. The free PDF download of NCERT Solutions facilitates easy access to valuable study materials, supporting students in their mathematical journey. Overall, these solutions encourage a deeper appreciation for patterns and shapes, fostering a solid foundation in mathematics and preparing students for more advanced concepts.

Other Study Material for CBSE Class 5 Maths Chapter 3

Chapter-Specific NCERT Solutions for Class 5 Maths

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