Question

if a equilateral triangle has been inscribed in a circle of radius 6 cm. find the area of shaded region.( shaded region is the part lying outside the triangle i.e circle - triangle)

Answer 1

area of circle= πr^2
circum radius R=( ABC)/4*area
shaded region=area of circle- area of triangle
equilateral triangle:
R=(a^3)/(4*√3/4*a^2)
6=a/√3
a(side)=6√3
area =√3/4*(6√3)*(6√3)



area of Circle=22/7*36



area of shaded=( 22/7*36)-(3√3)/4*36
=113.14-46.764
=66.38

Answer 2

see the diagram.

Let the equilateral triangle be  ABC. Let the center of circle be O.
Let AOD be the altitude of the triangle.

O is the centroid, circumcenter, orthocenter, incenter of triangle ABC.

AOD = √3/2 * AB

So   Radius = AO = 2/3 * AOD = AB /√3     (∵ centroid O is at 1/3 height on median)

So    AB = √3 * radius = 6√3 cm

Area of circle = π 6² = 36 π cm²
 Area of triangle ABC:  √3/4 * AB² = 27 √3 cm³

Shaded region = (36 π - 27√3) cm²