PK 2019 Q14
14. The diagram shows a shape made of arcs of three circles, each with radius \(R\). The centres of the circles lie on the same straight line, and the middle circle passes through the centres of the other two circles. What is the perimeter of the shape?
- A \(\dfrac{2\pi R\sqrt{3}}{3}\)
- B \(\dfrac{5\pi R}{3}\)
- C \(\dfrac{10\pi R}{3}\)
- D \(2\pi R\sqrt{3}\)
- E \(4\pi R\)
Pink Kangaroo 2019, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(\dfrac{10\pi R}{3}\)
By drawing radii and chords as shown, we can see that the triangles are equilateral and therefore each of the angles is 60°. Hence the left and right circles have each lost 120° (one third) of their circumferences, and the central circle has one-third \(\left(\tfrac{1}{6} + \tfrac{1}{6}\right)\) of its circumference remaining. Since \(1 - \tfrac{1}{3} + 1 - \tfrac{1}{3} + \tfrac{1}{3} = \tfrac{5}{3}\), the perimeter of the shape is \(\tfrac{5}{3} \times 2\pi R = \tfrac{10\pi R}{3}\).
Solution © UK Mathematics Trust, from the PK 2019 solutions.