IMC 2009 Q13
13. The diagram on the right shows a rectangle with sides of length
5 cm and 4 cm. All the arcs are quarter-circles of radius 2 cm.
What is the total shaded area in cm2?
- A \(12 - 2\pi\)
- B 8
- C \(8 + 2\pi\)
- D 10
- E \(20 - 4\pi\)
Intermediate Mathematical Challenge 2009, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A \(12 - 2\pi\)
If the shaded regions in the top-right and bottom-left corners of the diagram are moved as shown, the area of the shaded region in both the top half and bottom half of the diagram is now that of a 3 × 2 rectangle which has a quarter of a circle of radius 2 removed from it.
So the total shaded area is
2(3 × 2 − \(\tfrac{1}{4}\) × π × 22) cm2 = (12 − 2π) cm2.
Solution © UK Mathematics Trust, from the IMC 2009 solutions.