In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in figure. Find the area of the design (shaded region).

Let ABC be an equilateral triangle and let O be the circumcentre of the circumcircle of radius 32 cm.



Area of the circle \( = \ \pi r^2 \)

\( = \ \frac{22}{7} \ × \ (32)^2 \ = \ \frac{22528}{7} \) cm²

Area of \( ∆ \ ABC \) \(= \ 3 × \) Area of \(∆ \ BOC \)

\( = \ 3 \ × \ \frac{1}{2} \ × \ OB \ × \ OC \ × \ sinBOC \) \(= \ \frac{3}{2} \ × \ 32 \ × \ 32 \ × \ sin120° \)

\( = \ 3 \ × \ 16 \ × \ 32 \ × \ \frac{ \sqrt{3}}{2} \ = \ 768 \sqrt{3} \)cm²

\(\therefore \) Area of the design
(i.e., shaded region)

= Area of the circle - Area of ∆ ABC

\(= \ ( \frac{22528}{7} \ - \ 768 \sqrt{3}\)) cm²