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 \(5\pi:18\).

What do you think it means for the circle to “intercept a length of \(2x\) units on each side of the square”?

Could we draw a diagram to make it clearer?

If you’re still unsure, click to reveal how we’ve drawn it.

Circle with centre the same as that of the square, so that its diameter is slightly larger than the side of the square, so that a segment in the middle of each side of the square is enclosed within the circle.
The square and the circle

Can you spot a right-angled triangle in your diagram that will help you to find the radius of the circle?