IMC 2010 Q9
9. The perimeters of the three shapes shown are made up of straight lines and semi-circular arcs of diameter 2. They will fit snugly together as in a jigsaw.
What is the difference between the total perimeter of the three separate pieces and the perimeter of the shape formed when the three pieces fit together?
- A 0
- B \(2 + 2\pi\)
- C \(8 + 4\pi\)
- D \(22 + 2\pi\)
- E \(30 + 6\pi\)
Intermediate Mathematical Challenge 2010, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(8 + 4\pi\)
The difference in perimeters is the total length of the edges which are hidden when the pieces are fitted together. These are eight straight edges of length 1 and four semicircular arcs of radius 1.
So the required difference is 8 × 1 + 4(\(\tfrac{1}{2}\) × 2 × π × 1) = 8 + 4π.
Solution © UK Mathematics Trust, from the IMC 2010 solutions.