Question

Area of a sector is $\frac{\pi r^2\theta }{{360}^{\circ }}$

All the sides of a square are equal.

The figure below consists of a square, a quarter circle and a semicircle

(a) Find the area of the shaded portion.

(b) Find the perimeter of the shaded portion

• A. (a) $\frac{25\pi }{2}$
  (b) $(1+\pi )$
• B. (a) $\frac{23\pi }{2}$
  (b) $9(1+\pi )$
• C. (a) $\frac{25\pi }{2}$
  (b) $10(1+\pi )$
• D. none

Answer 1

Answer: (C)

(a) $\frac{25\pi }{2}$

(b) $10(1+\pi )$

(a)Area of square=$(side{)}^2={10}^2=100c{m}^2$

For small semicircle radius $r=\frac{AB}{2}=\frac{10}{2}=5cm$

Area of semi circle=$\frac{\pi .r^2}{2}=\frac{\pi \times 5^2}{2}=12.5\pi c{m}^2$

For quadrant radius r=10cm

Area of quadrant of circle=$\frac{\pi .R^2}{4}=\frac{\pi \times 100}{4}=25\pi c{m}^2$

Area of shaded region= Area Quadrant-Area of semicircle

=$25\pi -\frac{25\pi }{2}=\frac{25\pi }{2}c{m}^2$

(b) Perimeter of shaded portion=length of arc of small semicircle+length of arc of quadrant+side of square

=$\frac{2\pi .r}{2}+\frac{2.\pi .R}{4}+10=5\pi +5\pi +10=10(\pi +1)cm$