Question

Find the area of the shaded portion in the equilateral triangle with sides 6.
(assuming the central point of each arc is its corresponding vertex)

click the pic below

Answer 1

height of the triangle (using pythagoras):
h^2+3^2=6^2 --> h=3*sqrt(3)

TriangleArea = base*height/2:
6*3*sqrt(3)/2 = 9*sqrt(3)

We have 3 white sectors, each defined by a 60 degree angle and radius of 3 -
so all three sectors together cover the area of half a circle (180 degrees).

ThreeSectorsArea = Area of half a circle:
(pi*r^2)/2 = (9*pi)/2

The shaded area = TriangleArea - ThreeSectorsArea
= 9*sqrt(3) - 9*pi/2