Question

The side of a square is 10 cm. Find the area of circumscribed and inscribed circles.

Answer 1

$\huge \boxed {\underline{{ \mathrm{Solution:}}}} \\ \\ \sf \underline{ \underline{Given : }} \\ \rm Side \: of \: a \: Square = \boxed { \boxed{ \text{10 cm}}} \\ \\ \sf\underline { \underline{Explanation : }} \\ \sf When \: it \: is \: Circumscribed : \\ \rm The \: Diameter \: will \: be \: Square \: Side \\ \sf \therefore r = \boxed { \boxed{ \text{5 cm}}} \\ \\ \rm Area \: of \: Circle :\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \sf \pi r {}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \sf \frac{22}{7} \times 5 \times 5 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \boxed{ \boxed{ \sf 78.57 \: cm {}^{2}}} \\ \\ \\ \sf When \: it \: is \: Inscribed : \\ \rm T he \: Diameter \: will \: be \: its \: Diagonal \\ \therefore \rm r = \boxed{ \boxed{ \rm 5\sqrt{2} }} \\ \\ \rm Area \: of \: Circle : \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \sf \pi r {}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \sf \frac{22}{7} \times 5 \sqrt{2} \times 5 \sqrt{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \sf \boxed{ \boxed{ \sf 157.14 \: cm {}^{2} }}$

Answer 2

given the side of the square = 10cm

by Pythagoras theorem we get,

length of it's diagonal = √(10² + 10²)

since the diagonal of a square divides the square in 2 right angle triangles. and the diagonal hence become the hypotenuse of the triangle.

= √(100 + 100)

= √200

= 10√2cm

now, areas of the circumscribed and inscribed circles..

circumscribed circle are those circles where the circle touches all the veriticies of the given polygon. here it's circumscribed by the square. circles circumscribed by square have their diameter = diagonal of the square

therefore it's diameter = 10√2cm

➡ it's radius = 10√2/2

= 5√2cm

hence area of the circumscribed circle = πr²

= 22/7 × (5√2)²

= 22/7 × 50

= 1100/7

= 157.14cm² (approximately)

inscribed circles are those circles which are largest circle can be drawn in the given polygon. circle inscribed in a square have their diameter = side of the square

therefore it's diameter = 10cm

➡ it's radius = 10/2 = 5cm

hence, area of the inscribed circle = 22/7 × 5 × 5

= 22/7 × 25

= 550/7

= 78.57cm² (approximately)