IMC 2008 Q17
17. The shaded region is bounded by eight equal circles with centres at the corners and midpoints of the sides of a square. The perimeter of the square has length 8. What is the length of the perimeter of the shaded region?
- A \(\pi\)
- B \(2\pi\)
- C 8
- D \(3\pi\)
- E \(4\pi\)
Intermediate Mathematical Challenge 2008, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(3\pi\)
As the perimeter of the square has length 8, the square has side length 2. So the diameter of each of the circles is 1. The perimeter of the shaded region consists of four semi-circular arcs and four quarter-circle arcs, so it has length equal to three times the circumference of one circle, that is \(3\pi\).
Solution © UK Mathematics Trust, from the IMC 2008 solutions.