IMC 2012 Q19
19. The shaded region shown in the diagram is bounded by four arcs, each of the same radius as that of the surrounding circle. What fraction of the surrounding circle is shaded?
- A \(\dfrac{4}{\pi} - 1\)
- B \(1 - \dfrac{\pi}{4}\)
- C \(\dfrac{1}{2}\)
- D \(\dfrac{1}{3}\)
- E it depends on the radius of the circle
Intermediate Mathematical Challenge 2012, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A \(\dfrac{4}{\pi} - 1\)
The diagram shows the original diagram enclosed within a square of side 2\(r\), where \(r\) is the radius of the original circle. The unshaded area of the square consists of four quadrants (quarter circles) of radius \(r\). So the shaded area is \(4r^2 - \pi r^2\) \(= r^2(4 - \pi)\). Therefore the required fraction is
Solution © UK Mathematics Trust, from the IMC 2012 solutions. Longer versions with investigations: Solutions and investigations (PDF).