The perimeter of a semicircular plate is 108 cm, find its area.

Given:

Perimeter of the semicircular plate = 108 cm

Let r be the radius of the plate.

The perimeter of the semicircular plate includes the curved part (half the circumference of a circle) and the diameter. So,

⇒ πr + 2r = Perimeter

⇒ πr + 2r = 108

⇒ r(π + 2) = 108

⇒ r $\Big(\dfrac{22}{7} + 2\Big)$ = 108

⇒ r $\Big(\dfrac{22}{7} + \dfrac{14}{7}\Big)$ = 108

⇒ r $\Big(\dfrac{22 + 14}{7} \Big)$ = 108

⇒ r $\Big(\dfrac{36}{7}\Big)$ = 108

⇒ r = $\Big(\dfrac{7 \times 108}{36}\Big)$

⇒ r = 7 x 3 = 21 cm

Area of semicircular plate = $\dfrac{1}{2}$ πr²

$= \dfrac{1}{2} \times \dfrac{22}{7} \times 21^2\\[1em] = \dfrac{22}{14} \times 441\\[1em] = \dfrac{9,702}{14} \\[1em] = 693 \text{ cm}^2$

Hence, the area of the semicircular plate is 693 cm².