Question

A square is inscribed in a circle with diameter 10 cm. four semi-circles are then constructed outside it, with their diameters on each side of the square. find :

a.area of the square

b.area of the 4 semicircles

Answer 1

(a) Diagonal of the square = Diameter of the circle = 10cm

$\therefore$ Side of the square = $\sf{\frac{d}{ \sqrt{2} } = \frac{10}{ \sqrt{2} } cm}$

Area of the square = $\sf{ {( \frac{10}{ \sqrt{2} }cm )}^{2}}$

= $\sf{ \frac{100}{2} {cm}^{2} }$

= $\sf{50 {cm}^{2}}$

(b) Diameter of each semicircle = Side of the square = $\sf{ \frac{10}{ \sqrt{2} } cm}$

$\longrightarrow$ Radius of the semicircle = $\sf{ \frac{5}{ \sqrt{2} } cm}$

$\therefore$ Area of four semicircles

= $\sf{4 \times \frac{1}{2} \times \pi \times {( \frac{5}{ \sqrt{2} }) }^{2} {cm}^{2} }$

= $\sf{2 \times 3.14 \times \frac{25}{2} {cm}^{2} }$

= $\sf{78.5 {cm}^{2} }$