Let us study a Circle inscribed in Equilateral Triangle and Square

First let’s see the Circle in Equilateral Triangle

Circle in 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

(L – r) 2 =  r 2 + (t/2) 2

r2 = (√3t/2 – r)2 – (t/2)2

r2 = 3t2 /4 + r2 – √3tr – t2 /4

√3tr = t2 /2

t  = 2√3 r                &               r = t/ 2√3

AreasIn terms of radius rIn terms of side t
Area of equilateral
triangle ABC with side ‘t’
3 √3 r2√3 t2 /4
Area of circle with
radius ‘r’
∏ r2∏ t2 /12

Let us see Circle inscribed in Square

Circle in Square

s = r + r

s = 2r                 &               r = s/ 2

AreasIn terms of radius rIn terms of side t
Area of square ABCD with side ‘s’4 r2s2
Area of circle with
radius ‘r’
∏ r2∏ s2 /4
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