Question

4 in
6 in.
4 in

Find the aria of the shaded region

Answer 1

Check the picture below.

first off let's get the area of the "circular ring", and then let's add to that the area of the innermost circle with a radius of 6.

`\textit{area of a circular ring}\\\\ A=\pi (R^2 - r^2) ~~ \begin{cases} R=\stackrel{outer}{radius}\\ r=\stackrel{inner}{radius}\\[-0.5em] \hrulefill\\ R=14\\ r=10 \end{cases}\implies A=(14^2-10^2)\implies A=96\pi \\\\[-0.35em] ~\dotfill\\\\ \textit{area of a circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=6 \end{cases}\implies A=\pi (6)^2\implies A=36\pi \\\\[-0.35em] ~\dotfill\\\\ 96\pi ~~ + ~~36\pi \implies 132\pi ~~ \approx ~~ \text{\LARGE 414.69}~in^2`