ANSWERs for figure it out Class 8 Mathematics – NCERT (Ganita Prakash) Part 2 CH7  AREA CLASS 8 

Figure it Out Page number 150

1. Identify the missing side lengths

(i)

Given:

• Area = 35 in²

• One side = 7 in

$$\text{Other side} = \frac{35}{7} = 5 \text{ in}$$

Answer: Missing side = 5 in

(ii)

Given:

• Area = 28 in²

• One side = 4 in

$$\text{Other side} = \frac{28}{4} = 7 \text{ in}$$

Answer: Missing side = 7 in

(iii)

Given:

• Area = 21 in²

• One side = 3 in

$$\text{Other side} = \frac{21}{3} = 7 \text{ in}$$

Answer: Missing side = 7 in

(iv)

Given:

• Area = 29 m²

• One side = 4 m

$$\text{Other side} = \frac{29}{4} = 7.25 \text{ m}$$

Answer: Missing side = 7.25 m

(v)

Given:

• Area = 11 m²

• One side = ?

Multiple answers possible (activity-based).
One possible pair:

$$11=11\times 1$$

Answer:
Sides may be 11 m and 1 m

(vi)

Given:

• Area = 50 m²

Possible pairs:

• 10 m × 5 m

• 25 m × 2 m

Answer:
One possible set of sides: 10 m and 5 m

2. The figure shows a path (the shaded portion) laid around a rectangular park EFGH. (i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area. An example of a formula — Area of a rectangle = length × width. [Hint: There is a relation between the areas of EFGH, the path, and ABCD.]

Path around a rectangular park

A rectangular park EFGH is surrounded by a path (shaded region).

(i) Measurements needed to find the area of the path

To find the area of the path, we need:

• Length of the outer rectangle (ABCD)

• Breadth of the outer rectangle

• Length of the inner rectangle (EFGH)

• Breadth of the inner rectangle

Area of path=Area of outer rectangle−Area of inner rectangle\text{Area of path} = \text{Area of outer rectangle} - \text{Area of inner rectangle}

Let:

• Outer rectangle = 20 m × 14 m

• Inner park = 16 m × 10 m

$$\text{Area of outer rectangle} = 20 × 14 = 280 \text{ m}^2$$
$$\text{Area of inner park} = 16 × 10 = 160 \text{ m}^2$$
$$\boxed{\text{Area of path} = 280 - 160 = 120 \text{ m}^2}$$

(ii) If the width of the path along each side is given, can you find 

its area? If not, what other measurements do you need? Assign 

values of your choice to these measurements and find the area of 

the path. Give a formula for the area using these measurements. 

[Hint: Break the path into rectangles.]

(ii) If width of path is given

Suppose:

• Width of path = x m

• Inner park dimensions = l × b

Then:

• Outer length = $l + 2x$

• Outer breadth = $b + 2x$

$$\text{Area of path} = (l+2x)(b+2x) - lb$$

(iii) Does the area of the path change when the outer rectangle is moved while keeping the inner rectangular park EFGH inside it, as shown?

Answer: ❌ No

Reason:
Area depends only on dimensions, not on position.
As long as the inner park and width of the path remain the same, area remains unchanged.

3. The figure shows a plot with sides 14m and 12m, and with a crosspath. What other measurements do you need to find the area of the crosspath? Once you identify the lengths to be measured, assign some possible values of your choice and find the area of the path. Give a formula for the area based on the measurements you choose.

Cross path in a rectangular plot
Plot dimensions = 14 m × 12 m
To find the area of the cross path, we need:
Width of the horizontal path
Width of the vertical path

Let:

Width of each path = 2 m

Area of horizontal path:

$$14 × 2 = 28 \text{ m}^2$$

Area of vertical path:

$$12 × 2 = 24 \text{ m}^2$$

Overlapping square counted twice:

$$2 × 2 = 4 \text{ m}^2$$$$\text{Area of cross path} = 28 + 24 - 4 = \boxed{48 \text{ m}^2}$$

If width of each path = w:

$$\boxed{\text{Area of cross path} = lw + bw - w^2}$$

4. Find the area of the spiral tube shown in the figure. The tube has the same width throughout [Hint: There are different ways of finding the area. Here is one method.] What should be the length of the straight tube if it is to have the same area as the bent tube on the left?

Area of spiral tube

The tube has uniform width.

Break the spiral into rectangular strips of width 5 units.

Add the areas of all straight rectangular parts.

(Exact numerical answer depends on the given dimensions in the figure.)

Length of straight tube

To have the same area:

$$\text{Length of straight tube} = \frac{\text{Area of bent tube}}{\text{Width}}$$

​

5. In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons.

Doubling the side of a square

If original side = s
New side = 2s

Areas

• Original area = $s^2$

• New area = $(2s)^2 = 4s^2$

Increase in regions 1, 2, and 3

Each region’s area becomes 4 times its original area.

Reason:
Area is proportional to the square of the side.

6. Divide a square into 4 parts by drawing two perpendicular lines inside the square as shown in the figure. Rearrange the pieces to get a larger square, with a hole inside. You can try this activity by constructing the square using cardboard, thick chart paper, or similar materials.

Rearranging square pieces

• Square divided into 4 equal parts.

• Rearranged to form a larger square with a hole.

Conclusion:
✔️ Area remains the same
✔️ Shape changes, area does not

Figure it Out Page number 157-158

1. Find the areas of the following triangles:

2. Find the length of the altitude BY

3. Find the area of ∆SUB, given that it is isosceles, SE is perpendicular to UB, and the area of ∆SEB is 24 sq. units.

4. [Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area. 

5. [Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area. 

6. ABCD, BCEF, and BFGH are identical squares. 

(i) If the area of the red region is 49 sq. units, then what is the area of the blue region? 

(ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square?

7. If M and N are the midpoints of XY and XZ, what fraction of the area of ∆XYZ is the area of ∆XMN? [Hint: Join NY]

8. Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path.

Figure it Out page number 160

 1. Find the area of the quadrilateral ABCD given that AC = 22 cm, BM = 3 cm, DN = 3 cm, BM is perpendicular to AC, and DN is perpendicular to AC

2. Find the area of the shaded region given that ABCD is a rectangle

3. What measurements would you need to find the area of a regular hexagon?

4. What fraction of the total area of the rectangle is the area of the blue region?

5. Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral. One can derive special formulae to find the areas of a parallelogram, rhombus and trapezium.

Figure it Out Page number 162 - 163

1. Observe the parallelograms in the figure below. 

(i) What can we say about the areas of all these parallelograms?

 (ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?

2. Find the areas of the following parallelograms:

3. Find QN.

 4. Consider a rectangle and a parallelogram of the same sidelengths: 5 cm and 4 cm. Which has the greater area? [Hint: Imagine constructing them on the same base.]

5. Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods that you can think of?

6. [Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.

 7. [Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it? 
[Hint: Show that triangles ∆ADB and ∆ADC can be made into halves of a rectangle. Figure out how they should be assembled to get a rectangle. Use cut-outs if necessary.]

8. [Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection. 

9. Which has greater area — an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area — two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.

Figure it Out Page number 169-170

 1. Find the area of a rhombus whose diagonals are 20 cm and 15 cm. 

2. Give a method to convert a rectangle into a rhombus of equal area using dissection. 

 3. Find the areas of the following figures:

4. [Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.

 5. Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area —

Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH? [Hint: If ∆AHI ≅ ∆DGI and ∆BEJ ≅ ∆CFJ, then the trapezium and rectangle have equal areas.]

6. Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area 144 cm2.





7. A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.

8. ZYXW is a trapezium with ZY‖WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of ∆ZWB.

Areas in Real Life

What do you think is the area of an A4 sheet? Its sidelengths are 21 cm and 29.7 cm. Now find its area

What do you think is the area of the tabletop that you use at school or at home? You could perhaps try to visualise how many A4 sheets can fit on your table. The dimensions of furniture like tables and chairs are sometimes measured in inches (in) and feet (ft). 1 in = 2.54 cm 1 ft = 12 in

Express the following lengths in centimeters: (i) 5 in (ii) 7.4 in

Express the following lengths in inches: (i) 5.08 cm (ii) 11.43 cm

How many cm² is 1 in² ? So, 1 in² = 2.54² cm² = 6.4516 cm²

How many cm² is 10 in² ? 10 in² = 10 × 6.4516 cm² = 64.516 cm².

How many cm² is 10 in² ? 10 in² = 10 × 6.4516 cm² = 64.516 cm².

Convert 161.29 cm² to in². Every 6.4516 cm² gives an in². Hence, 161.29 cm² = 161.29 /6.4516 in² Evaluate the quotient.

What do you think is the area of your classroom? Areas of classroom, house, etc., are generally measured in ft² or m²

How many in² is 1 ft² ?

What do you think is the area of your school? Make an estimate and compare it with the actual data. Larger areas of land are also measured in acres. 1 acre = 43,560 ft2. Besides these units, different parts of India use different local units for measuring area, such as bigha, gaj, katha, dhur, cent, ankanam, etc. 

Find out the local unit of area measurement in your region.

What do you think is the area of your village/town/city? Make an estimate and compare it with the actual data. Larger areas are measured in km².

How many m² is a km²  ?

How many times is your village/town/city bigger than your school?

Find the city with the largest area in (i) India, and (ii) the world

Find the city with the smallest area in (i) India, and (ii) the world