+128460 First one.....the area of the inscribed equilateral triangle =
3(1/2)r^2 sin 120 = ( 3√3/4) r^2
The Height of the circumscribed equilateral triangle = 3r
And we can find the base as
tan 60 = 3r /(1/2) Base
√3 =6r / Base
Base = 6r/√3 = 2√3r
So....the area of the circumscribed eqilateral triangle is (1/2)Base* Height = (1/2)2√3r * 3r = 3√3r^2
So we have that
Area of circumscribed triangle - Area of inscribed triangle = 36
(3√3 - 3√3/4) r^2 = 36
3(√3 - √3/4) r^2 = 36
(√3 - √3/4)r^2 = 12
r^2 = 12 / ( √3 - √3/4 )
r = √ [ 12 / ( √3 - √3/4 ) ] ≈ 3.04 cm
