2D shapes and their properties

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

The diagram shows two semicircular arcs... What is the diameter of the shaded region?

The diagram shows two circles and four equal semi-circular arcs. The area of the inner shaded circle is 1. What is the area of the outer circle?

Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?

In this problem, we're going to find sets of letter shapes that go together.