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 |