Question

The circles have the same center. What is the area of the shaded region?

Answer 1

The area of the shaded region is 160.2 in² (to the nearest tenth).

Explanation:

The area of the shaded region can be calculated by subtracting the area of the inner circle from the area of the outer circle.

`\boxed{\begin{minipage}{4 cm}\underline{Area of a circle}\\\\$A=\pi r^2 $\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\\end{minipage}}`

If the circles have the same center, and the inner circle has a radius of 7 in, then the outer circle has a radius of:

`\implies r = 7 + 3 = 10\; \sf in`

Therefore, the area of the shaded region is:

`\begin{aligned}\sf Area_(shaded\;region)&=\sf Area_(outer\;circle)-Area_(inner\;circle)\\&=\pi \cdot 10^2 - \pi \cdot 7^2\\&=100\pi - 49 \pi\\&=51 \pi\\& = 160.221225...\\&=160.2\; \sf in^2\;(nearest\;tenth)\end{aligned}`

Answer 2

Answer: 160.14

Explanation: the inner circle has an area of 153.86 in.^2. The outer circle as a whole has an area of 314 in.^2. Subtract the area of the inner circle(153.86) from the area of the total of the bigger circle(314) to get the area of the shaded region of the circle.