Review question

# What's the ratio of the circle's area to the square's area? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8179

## Solution

A square has each side of length $6x$ units. A circle is drawn with its centre at the centre of the square to intercept a length of $2x$ units on each side of the square. Prove that the ratio of the area of the circle to the area of the square is $5\pi:18$.

The square and circle look like this:

We know that the area of the square is $6x\times 6x=36x^2$. To find the radius, $r$, of the circle we use the following triangle:

The hypotenuse of the triangle is $r$, one side has length $x$ and the other has length $3x$. Therefore, $r^2=x^2+(3x)^2=10x^2.$ So the area of the circle is $\pi r^2=10x^2\pi$.

So the ratio of the area of the circle to the area of the square is $10x^2\pi:36x^2$, which is equivalent to $5\pi:18$, as required.

You might have noticed that the $x$ doesn’t appear in our answer. Why is this? Could we have solved this without using $x$?