# Thread: Finding the length of the wire.

1. ## Finding the length of the wire.

A wire 360 in. long is cut into two pieces. One piece is formed into a square and the other into circle. If the two figures have the same area, what are the lengths of the two pieces of wire?(to the nearest tenth of an in.) help guys, i can't figure this out (((

2. You have a square and a circle, and they have the same area.

You should know that the perimeter of a square is $\displaystyle 4l$ and the circumference of a circle is $\displaystyle 2\pi r$.

When you add them, you have the length of the wire, so

$\displaystyle 4l + 2\pi r = 360$.

The area of a square is $\displaystyle l^2$ and the area of a circle is $\displaystyle \pi r^2$.

Since the areas are equal, that means

$\displaystyle l^2 = \pi r^2$

or $\displaystyle l = \sqrt{\pi}r$.

Substituting this back into the length of the wire formula gives

$\displaystyle 4\sqrt{\pi}r + 2\pi r = 360$

$\displaystyle 2\sqrt{\pi}r(2 + \sqrt{\pi}) = 360$

$\displaystyle \sqrt{\pi}r(2 + \sqrt{\pi}) = 180$

$\displaystyle r = \frac{180}{\sqrt{\pi}(2 + \sqrt{\pi})}$.

Since $\displaystyle l = \sqrt{\pi} r$ that means

$\displaystyle l = \sqrt{\pi}\left[\frac{180}{\sqrt{\pi}(2 + \sqrt{\pi})}\right]$

$\displaystyle l = \frac{180}{2 + \sqrt{\pi}}$.

So the two pieces of wire are...

$\displaystyle P = 4l$

$\displaystyle = 4\left(\frac{180}{2 + \sqrt{\pi}}\right)$

$\displaystyle = \frac{720}{2 + \sqrt{\pi}}$

and

$\displaystyle C = 2\pi r$

$\displaystyle = 2\pi \left[\frac{180}{\sqrt{\pi}(2 + \sqrt{\pi})}\right]$

$\displaystyle = \frac{360\pi}{2\sqrt{\pi} + \pi}$.

Of course, all units are in inches.