Question

What is the perimeter of a square which has the same area as a circle

with circumference of 471?

Answer 1

`\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=471 \end{cases}\implies 471=2\pi r\implies \cfrac{471}{2\pi }=r \\\\\\ \textit{area of a circle}\\\\ A=\pi r^2\qquad \implies A=\pi \left( \cfrac{471}{2\pi } \right)^2\implies A=\pi\cdot \cfrac{471^2}{4\pi^2} \implies A=\cfrac{471^2}{4\pi }`

`\textit{area of a square}\\\\ A=s^2~~ \begin{cases} s=side\\[-0.5em] \hrulefill\\ A=(471^2)/(4\pi ) \end{cases}\implies \cfrac{471^2}{4\pi }=s^2\implies \sqrt{\cfrac{471^2}{4\pi }}=s \implies \cfrac{471}{2√(\pi )}=s \\\\\\ \stackrel{\textit{perimeter of a square}}{4s\implies 4\left( \cfrac{471}{2√(\pi )} \right)}\implies \cfrac{942}{√(\pi )}~~ \approx~~531.47`