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
BA^{2} = AP^{2} + BP^{2}
t^{2} = L^{2} + (t/2)^{2}
L^{2} = 3t^{2} /4
L = √3 t/2
Again by Pythagoras theorem in triangle BOP
BO^{2} = OP^{2} + BP^{2}
r^{2} = (L – r) ^{2} + (t/2)^{ 2}
r^{2} = (√3t/2 – r)^{2} + (t/2)^{2}
r^{2} = 3t^{2} /4 + r^{2} – √3tr + t^{2} /4
√3tr = t^{2}
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 r^{2} /4 | √3 t^{2} /4 |
Area of circle with radius ‘r’ | ∏ r^{2} | ∏ t^{2} /3 |