Let us study a Circle inscribed in Equilateral Triangle and Square

First let’s see the Circle in 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}

(L – r)^{ 2} = 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} /2

t = 2√3 r & r = t/ 2√3

Areas | In terms of radius r | In terms of side t |

Area of equilateral triangle ABC with side ‘t’ | 3 √3 r^{2} | √3 t^{2} /4 |

Area of circle with radius ‘r’ | ∏ r^{2} | ∏ t^{2} /12 |

Let us see Circle inscribed in Square

s = r + r

s = 2r & r = s/ 2

Areas | In terms of radius r | In terms of side t |

Area of square ABCD with side ‘s’ | 4 r^{2} | s^{2} |

Area of circle with radius ‘r’ | ∏ r^{2} | ∏ s^{2} /4 |