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NCERT Solutions Ganita Prakash Chapter 6 Figure it Out — — mixed practice on perimeter and area

Book page 149 Updated on2026-09-05

Q1.
Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: 5 m × 10 m and 2 m × 7 m.
Answer

Step 1 — add the two areas.

5 m × 10 m = 50 sq m
2 m × 7 m = 14 sq m
Total = 50 + 14 = 64 sq m

Step 2 — find a rectangle with this area. Any pair of numbers multiplying to 64 will do:

DimensionsAreaPerimeter
8 m × 8 m (a square)64 sq m32 m
16 m × 4 m64 sq m40 m
32 m × 2 m64 sq m68 m
64 m × 1 m64 sq m130 m

Answer: 8 m × 8 m is the neatest choice (16 m × 4 m, 32 m × 2 m and 64 m × 1 m are equally correct).

Q2.
The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden.
Answer
Area = length × width
1000 = 50 × width
width = 1000 ÷ 50 = 20 m

Answer: the garden is 20 m wide.

Check it yourself: 50 × 20 = 1000 ✔   Its perimeter, if you need it, is 2 × (50 + 20) = 140 m.
Q3.
The floor of a room is 5 m long and 4 m wide. A square carpet whose sides are 3 m in length is laid on the floor. Find the area that is not carpeted.
Answer
Area of the floor = 5 m × 4 m = 20 sq m
Area of the square carpet = 3 m × 3 m = 9 sq m
Area not carpeted = 20 − 9 = 11 sq m

Answer: 11 square metres of the floor is left uncovered.

Why we subtract: the carpet hides part of the floor. Whatever is left over is “whole floor minus carpet”. This same subtraction idea works for a garden with a pond, a wall with a window, or a field with a hut.
Q4.
Four flower beds having sides 2 m long and 1 m wide are dug at the four corners of a garden that is 15 m long and 12 m wide. How much area is now available for laying down a lawn?
Answer
Area of the garden = 15 m × 12 m = 180 sq m
Area of one flower bed = 2 m × 1 m = 2 sq m
Area of four flower beds = 4 × 2 = 8 sq m
Area left for the lawn = 180 − 8 = 172 sq m

Answer: 172 square metres are available for the lawn.

Careful: the beds are rectangles 2 m × 1 m, not squares. Each covers only 2 sq m.
Q5.
Shape A has an area of 18 square units and Shape B has an area of 20 square units. Shape A has a longer perimeter than Shape B. Draw two such shapes satisfying the given conditions.
Answer

We need the smaller shape to have the longer boundary — which is possible because area and perimeter are independent. Make A long and thin, and B square-like.

ShapeDraw it asAreaPerimeter
Aa 2 × 9 rectangle18 sq units2 × (2 + 9) = 22 units
Ba 4 × 5 rectangle20 sq units2 × (4 + 5) = 18 units
Area: 18 < 20 ✔     Perimeter: 22 > 18 ✔

Other correct pairs: A = 1 × 18 (perimeter 38) with B = 4 × 5 (perimeter 18), or A = 1 × 18 with B = 2 × 10 (perimeter 24).

The idea being tested: a bigger area does not force a bigger perimeter. Stretching a shape thin makes the boundary long while the region stays small.
Q6.
On a page in your book, draw a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border?
Answer

First measure your page, then take off the margins.

Length of the border rectangle = page length − 1 cm − 1 cm = page length − 2 cm
Breadth of the border rectangle = page width − 1.5 cm − 1.5 cm = page width − 3 cm

Worked example for a page 28 cm long and 21 cm wide:

Border rectangle = (28 − 2) cm × (21 − 3) cm = 26 cm × 18 cm
Perimeter = 2 × (26 + 18) = 2 × 44 = 88 cm

A neat short cut:

Perimeter of the border = 2 × (L − 2 + B − 3) = 2 × (L + B) − 10
= perimeter of the page − 10 cm

So whatever the size of your page, the border's perimeter is always 10 cm less than the perimeter of the page. Measure your own book page and use this to get your answer.

Q7.
Draw a rectangle of size 12 units × 8 units. Draw another rectangle inside it, without touching the outer rectangle that occupies exactly half the area.
Answer
Area of the outer rectangle = 12 × 8 = 96 square units
Half of it = 96 ÷ 2 = 48 square units

Now choose an inner rectangle of area 48 that is small enough to leave a gap all round. Since it must not touch the outer rectangle, its length must be less than 12 and its width less than 8.

Inner rectangleAreaFits without touching?
8 × 648 sq unitsYes — leaves 2 units at left and right, 1 unit at top and bottom
6 × 848 sq unitsNo — 8 equals the full width
10 × 4.848 sq unitsYes, but the sides are not whole numbers

Answer: draw an 8 units × 6 units rectangle in the middle. Leave 2 units of space on the left and right and 1 unit at the top and bottom, so it nowhere touches the outer rectangle.

Q8.
A square piece of paper is folded in half. The square is then cut into two rectangles along the fold. Regardless of the size of the square, one of the following statements is always true. Which statement is true here? a. The area of each rectangle is larger than the area of the square. b. The perimeter of the square is greater than the perimeters of both the rectangles added together. c. The perimeters of both the rectangles added together is always 1½ times the perimeter of the square. d. The area of the square is always three times as large as the areas of both rectangles added together.
Answer

Statement (c) is the true one.

Take a square of side s. Folding and cutting gives two rectangles, each s long and s/2 wide. Try it with a square of side 8 cm:

Square (side 8 cm)Each rectangle (8 cm × 4 cm)Both rectangles
Area64 sq cm32 sq cm64 sq cm
Perimeter32 cm24 cm48 cm
48 ÷ 32 = 1½ ✔    so statement (c) holds.

Why the others fail:

  • a — each rectangle is only half the square (32 sq cm against 64 sq cm), so it is smaller, not larger.
  • b — the square's perimeter is 32 cm and the two rectangles together give 48 cm, which is more, not less. (Cutting creates two new edges.)
  • d — the two rectangles together have exactly the same area as the square, not one third of it.
The general proof: square side s → perimeter 4s. Each rectangle is s × s/2 → perimeter 2(s + s/2) = 3s. Two of them give 6s, and 6s = 1½ × 4s. It works for every square, which is what the question asks for.
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