NCERT Solutions Ganita Prakash (Part 2) Chapter 7 Why Can't Perimeter be a Measure of Area? — In-text Questions
Book page 150 Updated on2026-09-05
Q1.
What is the area of each triangle in this rectangle?
Answer
14 cm² each.
Area of the rectangle = 7 × 4 = 28 cm²
The diagonal cuts it into two congruent triangles
Area of each triangle = ½ × 7 × 4 = 14 cm²
Why it happens: a diagonal of a rectangle splits it into two triangles that fit exactly on each other (turn one through half a turn about the centre of the rectangle). Congruent figures have equal areas, and the two areas add up to 28 cm², so each must be 14 cm².
Q2.
Why do we count the number of unit squares to assign measures for area? Couldn't we have just used the perimeter of a region, i.e., the length of its boundary as a measure of its area?
Answer
No. Perimeter measures the boundary; area measures the region enclosed. They are different quantities, and one does not determine the other.
A unit square is the natural yardstick for a region because copies of it can be packed to fill up a region without gaps or overlaps. Counting how many fit tells us exactly "how much surface" there is.
Why it happens: perimeter is a length (measured in cm), area is a two-dimensional measure (measured in cm²). A rangoli of perimeter 22 cm might use 28 cm² of powder or 24 cm² of powder — the boundary length simply does not decide the answer.
Q3.
If two regions have the same perimeter, can't we conclude that they have the same area? Or, if one region has a larger perimeter than another region, can't we conclude that it also has a larger area?
Answer
No to both.
Rectangle
Perimeter
Area
7 cm × 4 cm
2(7 + 4) = 22 cm
28 cm²
8 cm × 3 cm
2(8 + 3) = 22 cm
24 cm²
10 cm × 1 cm
2(10 + 1) = 22 cm
10 cm²
Same perimeter, three different areas. So equal perimeters do not force equal areas.
Why it happens: with the perimeter fixed at 22 cm, the two sidelengths must add to 11 cm, but the product can be anything from just above 0 up to 5.5 × 5.5 = 30.25 cm². Fixing a sum does not fix a product.
Q4.
Find two rectangles that are examples of such regions. If needed, use a grid paper (given at the end of the book) for this.
Answer
We need Region 1 with the larger perimeter but the smaller area.
Region
Rectangle
Perimeter
Area
Region 1
12 cm × 1 cm
26 cm
12 cm²
Region 2
4 cm × 4 cm
16 cm
16 cm²
Perimeter of Region 1 = 26 cm > 16 cm = Perimeter of Region 2
Area of Region 1 = 12 cm² < 16 cm² = Area of Region 2 ✓
Why it happens: a long thin rectangle spends almost all of its boundary on the two long sides while enclosing very little. Stretching a rectangle out increases its perimeter but squeezes its area down.
Try This: on grid paper draw a 20 × 1 rectangle and a 5 × 5 square. The thin one has perimeter 42 units and area 20 squares; the square has perimeter 20 units and area 25 squares.
Q5.
Also give an example of two regions of other shapes, where the region with the larger perimeter has the smaller area! This property should be visually clear in your example. [Math Talk]
Answer
Take a square of side 6 cm and cut deep, thin notches into it, like the teeth of a comb.
Every extra slit adds a lot of boundary but removes area.
Region 2 (the plain square): perimeter 24 cm, area 36 cm².
Region 1 (the comb): each slit is 5 cm deep and very thin. Cutting it away removes only a sliver of area, but adds about 5 + 5 = 10 cm of new boundary. With 6 slits the perimeter grows past 80 cm while the area drops below 36 cm².
Why it happens: a slit of depth d and width w takes away area dw — which is tiny when w is tiny — but it adds boundary of length about 2d, which does not shrink at all as w shrinks. So perimeter can be made as large as we like while area only goes down.