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.

An equilateral triangle in a circle

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

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