Let the radius of the orange circle = r
So......its area = pi * r^2
We can find the side length,s, of the equilateral triangle thusly
tan (30°) = r / [(1/2)s]
1/ √3 = r / [ (1/2)s ]
(1/ √3) ( 1/2)s = r
s = (2√3) r = [ √12 ] r
And the area of the equilateral triangle is ( √3/ 4 ) ([√12]r)^2 = 3√3 r^2
So....the blue area inside the equilateral triangle =
[area of equailateral triangle - area of small circle ] / 3
[√3 r^2 - (1/3) pi r^2 ] = r^2 [ √3 - pi/3] (1)
The radius of the larger circle can be found as
√[ [√3r ]^2 + r^2 ] = √ [ 3r^2 + r^2 ] = 2r
So....the area of the larger circle = pi (2r)^2 = 4pi r^2
So the area between the side of the equilateral triangle and the larger circle is
[Area of larger circle - area of equilateral triangle ] / 3 =
[ 4pi r^2 - 3√3r^2 ] / 3 = r^2 [ (4/3)pi - √3] (2)
So the sum of (1) and (2) is the sum of the blue areas
r^2 [ √3 - pi/3 + (4/3) pi - √3 ] = pi r^2
So.....the orange and blue areas are equal !!!!