In given Fig. 6.50, four semicircles have been drawn within the given square whose side is 2 units. Class 9

In given Fig. 6.50, four semicircles have been drawn within the given square whose side is 2 units. Class 9

Question 1.

In given Fig. 6.50, four semicircles have been drawn within the given square whose side is 2 units. The centres of these semicircles are the midpoints of the sides. They create a 4-petalled flower (shown in blue). Find the perimeter and the area of this flower. Class 9

Solution:

i. Perimeter:

Given : Side of square = 2 units

Radius of each arc = $\frac{1}{2}$ × side = 1 unit

∵ Each petal is made of 2 quarter circles

∴ Length of arc of one petal = 2 × $\frac{1}{4}$ × 2πr

= πr

= $\frac{22}{7}$ × 1 = $\frac{22}{7}$

There are total 4 petals.

Total perimeter of petals = 4 × $\frac{22}{7}$ = $\frac{88}{7}$ units

∴ Total perimeter of petals = 12.57 units

ii. Area :

Given:

Side of square = 2 units

∴ Radius of semicircle = 1 unit

∵ Area of one semicircle = $\frac{1}{2}$ πr²

$=\frac{1}{2} \pi(1)^2=\frac{\pi}{2}$

∴ Area of 4 semicircles = $4 \times \frac{\pi}{2}=2 \pi$

∵ When adding all semicircles, the flower region is counted twice

The whole square is counted once

∴ Area of flower = Area of 4 semicircles - Area of square

∴ Area of flower = 2π - (2)²

$\begin{aligned} & =2 \times \frac{22}{7}-4 \\ & =\frac{44}{7}-4=\frac{16}{7} \\ & \approx 2.29\end{aligned}$

∴ Area of the flower = 2.29 sq. units