AC and BD are two perpendicular diameters of a circle ABCD. Given that the area of the shaded portion is 308 cm², calculate :

(i) the length of AC; and

(ii) the circumference of the circle.

Given,

Shaded region = 308 cm²

Area of circle = πr²

Two perpendicular diameters divide the circle into 4 quadrants.

Area of each quadrant = $\dfrac{1}{4}$ Area of circle

Since, 2 quadrants are shaded. Thus,

$\therefore \text{Shaded area} = 2 \times \dfrac{1}{4} \times \text{Area of circle} \\[1em] \Rightarrow 308 = \dfrac{1}{2} \times \text{Area of circle} \\[1em] \Rightarrow \text{Area of circle} = 616 \\[1em] \Rightarrow πr^2 = 616 \\[1em] \Rightarrow \dfrac{22}{7} r^2 = 616 \\[1em] \Rightarrow r^2 = \dfrac{616 × 7}{22} \\[1em] \Rightarrow r^2 = \dfrac{4312}{22} \\[1em] \Rightarrow r^2 = 196 \\[1em] \Rightarrow r = \sqrt{196} = 14 \text{ cm}.$

Diameter of circle = 2r = 28 cm.

(i) Length of AC = 28 cm.

Hence, length of AC = 28 cm.

(ii) Circumference of circle = 2πr

= 2 × $\dfrac{22}{7}$ × 14

= 2 × 22 × 2 = 88 cm.

Hence, circumference of circle = 88 cm.