Question

three semi circles have diameter of 3 cm a circle of diameter is 4.5 cm and a semi circle radius of 4.5cm​

Answer 1

Given :

• Three semi circles have Diameter of 3cm.

• circle diameter =4.5cm

•Semi circle radius =4.5cm

To Find :

• Required Area =?

Solution:

$\\ \implies\sf{Area \: of \: circle \: with \red{\bf{d=4.5cm}} \: \to \: (A_{1})=\dfrac{\pi}{4}(4.5)^{2}}$

$\\ \implies\sf{A_{1}=\dfrac{\pi}{4}(4.5)^{2}}$

$\\ \implies\boxed{\sf{A_{1}=\dfrac{\pi}{4}(4.5)^{2}}......(1)}$

$\\ \implies\sf{Area \: of \: 2 \: semicircle \: of \red{\bf{d=3cm}} \to \: (A_{2})=\dfrac{\pi}{4}(3)^{2}}$

$\\ \implies\sf{A_{2}=\dfrac{\pi}{4}(3)^{2}}$

$\\ \implies\boxed{\sf{A_{2}=\dfrac{\pi}{4}(3)^{2}}......(2)}$

$\\ \implies\sf{Area \: of \: semicircle \: of \: \red{\bf{r=4.5cm}} \to \: (A_{3})=\dfrac{1}{2}\dfrac{\pi}{4}(9)^{2}}$

$\\ \implies\sf{A_{3}=\dfrac{1}{2}\dfrac{\pi}{4}(9)^{2}}$

$\\ \implies\boxed{\sf{A_{3}=\dfrac{1}{2} \dfrac{\pi}{4}(9)^{2}}.....(3)}$

$\\ \implies\sf{Area \: of \: below \: shaded \: semicircle \: \red{\bf{d=3}} \: \to \:( A_{4})=\dfrac{1}{2}\dfrac{\pi}{4}(3)^{2}}$

$\\ \implies\sf{(A_{4})=\dfrac{1}{2}\dfrac{\pi}{4}(3)^{2}}$

$\\ \implies\boxed{\sf{(A_{4})=\dfrac{1}{2}\dfrac{\pi}{4}(3)^{2}}....(4)}$

$\\ \implies\sf{Required \: Area=(A_{3}-A_{1}-A_{2})+A_{4}}$

$\\ \implies \sf \: required \: area = \frac{1}{2} \frac{\pi}{4} {(9)}^{2} + \frac{1}{2} \frac{\pi}{4} {(3)}^{2} - \frac{\pi}{4} \times 4.5 - \frac{\pi}{4} {(3)}^{2} \\ \\ \\ \implies \sf \: \: \frac{\pi}{4} \bigg \{ \frac{81}{2} + \frac{9}{2} - 20.25 - 9 \bigg \} \\ \\ \\ \implies \sf \: \frac{ \pi}{4} \times 15.75 \\ \\ \\ \implies \sf 12.37 {cm}^{2}$

$\\ \red\bigstar\boxed{\bf{Required \: Area =\green{\bf{12.37cm²}}}}$