Question

the shaded region ABCD shows the space enclosed by two concentric circles with Centre O and the angle at the centre is 75 degrees if the radii of the circle are 21 cm and 42 cm find the area of the shaded region and the perimeter of ABCD

Answer 1

Recall:

$\text{Area} = \dfrac{\theta}{360} \times \pi r^2$

$\text{Arc Length} = \dfrac{\theta}{360} \times 2\pi r$

Find the area of the inner sector:

$\text{Area} = \dfrac{\theta}{360} \times \pi r^2$

$\text{Area} = \dfrac{75}{360} \times \pi (21)^2$

$\text{Area = }288.75 \text{ cm}^2$

Find the area of the outer sector:

$\text{Area} = \dfrac{\theta}{360} \times \pi r^2$

$\text{Area} = \dfrac{75}{360} \times \pi (42)^2$

$\text{Area = }1155 \text{ cm}^2$

Find area of the shaded region:

$\text{Shaded Area = }1155 - 288.75$

$\text{Shaded Area = }866.25 \text{ cm}^2$

Find length of arc AD:

$\text{Arc Length} = \dfrac{\theta}{360} \times 2\pi r$

$\text{Arc Length} = \dfrac{75}{360} \times 2\pi (21)$

$\text{Arc Length} = 27.5 \text{ cm}$

Find length of arc BC:

$\text{Arc Length} = \dfrac{\theta}{360} \times 2\pi r$

$\text{Arc Length} = \dfrac{75}{360} \times 2\pi (42)$

$\text{Arc Length} = 55 \text{ cm}$

Find the length of CD and AB:

$CD = 42 - 21$

$CD = 21 \text{ cm}$

$AB = CD$

$AB = 21 \text{ cm}$

Find the perimeter of the shaded region:

$\text{perimeter }= 27.5 + 55 + 21 + 21$

$\text{perimeter }= 124.5 \text { cm}$

Answer: Area = 866.25 cm², Perimeter = 124.5 cm