Question

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 ??

Answer 1

`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=4s` => then:
`s=P/4=L/4` => next the area of square we call `A_1`is:
`A_1`=`s^2=(L/4)^2=L^2/16`

Now for the circle we get:
`C=2*pi*r=L` => then the radius is:
`r=L/(2pi)` => the area of circle we call `A_2` is:
`A_2`=`pir^2=pi*[L/(2pi)]^2=L^2/(4pi)`

Finally the ratio of area of square divided by the area of the circle is:
`A_1/A_2` `=[L^2/(16)]/[L^2/(4pi)]=pi/4`

Answer 2

The ratio is `=pi/4`

Explanation:

The circumference of the circle is

`2pir=L`

The radius is `r=L/(2pi)`

The area of the circle is

`A_c=pir^2=pi*(L/(2pi))^2=piL^2/(4pi^2)`

`=L^2/(4pi)`

The length of each side of the square is

`l=L/4`

The area of the square is

`A_s=l^2=(L/4)^2=L^2/16`

The ratio is

`A_s/A_c=(L^2/16)/(L^2/(4pi))=pi/4`