Circumscribed Triangle

The triangle which is made in a circle such that all three vertices of a triangle touches circle is called Circumscribed Triangle.

Let us see the an example of Circumscribed equilateral triangle.

By Pythagoras theorem in triangle BAP

BA² = AP² + BP²

t² =  L² + (t/2)²    

L² =  3t² /4

L  =  √3 t/2

Again by Pythagoras theorem in triangle BOP

BO² = OP² + BP²

r² =  (L – r) ² + (t/2) ²

r² = (√3t/2 – r)² + (t/2)²

r² = 3t² /4 + r² – √3tr + t² /4

√3tr = t²

t  = √3 r                &               r = t/ √3

Areas	In terms of radius r	In terms of side t
Area of equilateral triangle ABC with side ‘t’	3 √3 r2 /4	√3 t2 /4
Area of circle with radius ‘r’	∏ r2	∏ t2 /3