Question

A wire$360\mathrm{in}$long is cut into two pieces. One piece is formed into a square, and the other is formed into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire?

Short answer

The length of wire made into a square is 190.9 in.

The length of wire made into a circle is 169.1 in.

Step-by-step solution

Step-1. Setup the model.

Circumference of the circle is$2\pi r.$

The perimeter of the square is 4l where l is the side.

The length of the wire is$360\mathrm{in}.$

It is given that the area of the circle and square are the same. So the radius is

$\begin{array}{l}\pi r^2=l^2\\ r^2=\frac{l^2}{\pi }\\ r=\sqrt{\frac{l^2}{\pi }}\\ r=\frac{l}{\sqrt{\pi }}\end{array}$

Step-2. Solve for l.

Circumference of the circular wire$2\pi r$ plus the perimeter of the square wire 4l is equal to360.

Thus,

$\begin{array}{c}2\pi r+4l=360\\ 2\pi \frac{l}{\sqrt{\pi }}+4l=360\\ 2l\sqrt{\pi }+4l=360\\ l(2\sqrt{3.14}+4)=360\\ l=\frac{360}{4+2\sqrt{3.14}}\\ l\approx 47.7\end{array}$

Step-3. Determine the length used.

The length of wire uses to make the square is$4\times 47.7=190.9\text{in.}$

The length of wire uses to make a circle is$360\text{in}-190.8\text{in}=169.1\text{in.}$