Question

A semicircle is drawn on each side of a square, as shown. The square has sides of length
2
π
.



What is the area of the resulting shape?

Answer 1

Given:

Length of each side of the square= 2π

To find:

Area of the resulting shape.

Solution:

Area of square =

${side}^{2}$

Therefore, area of square =

$( {2\pi})^{2}$

$= 4 {\pi}^{2}$

Diameter of semi circle = 2π

Radius of semi circle = π

Area of semi circle= $\frac{\pi r^{2} }{2}$

Here r= π

Area of 4 semi circles

$= 2\pi {r}^{2}$

$= 2\pi {\pi}^{2}$

$= 2 {\pi}^{3}$

Total area of the resulting shape= area of square + area of 4 semi circles

$4 {\pi}^{2} + 2 {\pi}^{3}$

Therefore the area of the resulting shape is $2\pi ^{2} (2+\pi )$ square units.

Answer 2

Given :  A semicircle is drawn on each side of a square

The square has sides of length 2π

To Find : area of the resulting shape

Solution:

Side of square =  2π

Area of Square = (2π )² = 4π²

Side of Square = Diameter of semicircle = 2π

=> Radius =  2π /2 = π

Area of Semicircle = (1/2)πr²

= (1/2)ππ²

= (1/2)π³

area of 4 Semicircles  = 4 ( (1/2)π³)  = 2π³

area of the resulting shape = 2π³ +  4π²

= 2π²(π + 2)

2π²(π + 2) sq units is the area of resulting shape

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