#1**+2 **

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

CPhill Apr 14, 2019