Question

What is the perimeter of a square which has the same area as a circle with circumference of 4pi?

Answer 1

`\textit{circumference of a circle}\\\\ C=2\pi r~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=4\pi \end{cases}\implies 4\pi =2\pi r\implies \cfrac{4\pi }{2\pi }=r\implies \boxed{2=r} \\\\[-0.35em] ~\dotfill\\\\ \textit{area of a circle}\\\\ A=\pi r^2\qquad \implies \stackrel{\textit{we know the circle has a radius of 2}}{A=\pi (2)^2\implies A=4\pi }`

so, since that's the area of that circle, that must also be the area of our square, we also know that say "s" being a side of a square, its area is simply s*s or s², so we can say that

`s^2=4\pi \implies s=√(4\pi )\implies s=2√(\pi )~~\leftarrow \textit{length of a side of the square} \\\\\\ \stackrel{\textit{since the perimeter of it is just 4 sides added up}}{2√(\pi )+2√(\pi )+2√(\pi )+2√(\pi )}\implies \blacktriangleright 8√(\pi ) \blacktriangleleft`