# A wire of length L can be shaped in to a circle or a square the ratio of area of a square divided by the area of circle is ??

Jul 6, 2018

$\frac{\pi}{4}$

#### Explanation:

Wire of length = L, shaped into a square then the perimeter of square is:
$P = L$ => perimeter of square is 4 times the length of a side:
$P = 4 s$ => then:
$s = \frac{P}{4} = \frac{L}{4}$ => next the area of square we call ${A}_{1}$is:
${A}_{1}$=${s}^{2} = {\left(\frac{L}{4}\right)}^{2} = {L}^{2} / 16$

Now for the circle we get:
$C = 2 \cdot \pi \cdot r = L$ => then the radius is:
$r = \frac{L}{2 \pi}$ => the area of circle we call ${A}_{2}$ is:
${A}_{2}$=$\pi {r}^{2} = \pi \cdot {\left[\frac{L}{2 \pi}\right]}^{2} = {L}^{2} / \left(4 \pi\right)$

Finally the ratio of area of square divided by the area of the circle is:
${A}_{1} / {A}_{2}$$= \frac{{L}^{2} / \left(16\right)}{{L}^{2} / \left(4 \pi\right)} = \frac{\pi}{4}$

Jul 6, 2018

The ratio is $= \frac{\pi}{4}$

#### Explanation:

The circumference of the circle is

$2 \pi r = L$

The radius is $r = \frac{L}{2 \pi}$

The area of the circle is

${A}_{c} = \pi {r}^{2} = \pi \cdot {\left(\frac{L}{2 \pi}\right)}^{2} = \pi {L}^{2} / \left(4 {\pi}^{2}\right)$

$= {L}^{2} / \left(4 \pi\right)$

The length of each side of the square is

$l = \frac{L}{4}$

The area of the square is

${A}_{s} = {l}^{2} = {\left(\frac{L}{4}\right)}^{2} = {L}^{2} / 16$

The ratio is

${A}_{s} / {A}_{c} = \frac{{L}^{2} / 16}{{L}^{2} / \left(4 \pi\right)} = \frac{\pi}{4}$