Question

an equilateral triangle is inscribed in an circle of radius 14cm. Find the area between the circle and the triangle​

Answer 1

for equilateral triangle

Answer 2

Area between Circle and Equilateral Triangle = 361.12 sq cm

Let O be the centre of circle and ΔABC be inscribed in the circle.

As ΔABC is an equilateral triangle, ∠ABC = 60°.

Thus, angle subtended at centre = 120°

As, the perpendicular drawn from the centre to the side of an inscribed triangle bisects the side, Let OD bisect side AC at D.

∴ Let AD = CD = x

Join OA.

In ΔOAD,

∠AOD = 60°.

∴ sin60 = $\frac{AD}{OA}$

But OA = Radius of circle = 14 cm

∴ AD = 14 × sin60

∴AD = 12.12 cm

∴ Side AC = 2 × AD

∴ Side AC = 24.25 cm

Area of Equilateral Triangle ABC = $\frac{root3}{4}$ × side²

∴ Area of ΔABC =  $\frac{root3}{4}$ × 24.25²

∴ Area of ΔABC = 254.63 sq cm

Area of Circle = π × R²

where R = Radius of Circle

∴ Area of Circle = π × 14²

∴ Area of Circle = 615.75 sq cm

Area between triangle and circle = Area of circle - Area of ΔABC

= 615.75 - 254.63

= 361.12 sq cm