IMC 2007 Q21
21. The diagram shows two circles and four equal semi-circular arcs.
The area of the inner shaded circle is 1.
What is the area of the outer circle?
- A \(\sqrt{2}\)
- B 2
- C \(1 + \sqrt{2}\)
- D \(\dfrac{\pi}{2}\)
- E \(\dfrac{9}{4}\)
Intermediate Mathematical Challenge 2007, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 2
From the symmetry of the figure, the two circles must be concentric. Let their centre be \(O\). Let the radius of the semicircles be \(r\). Then the radius of the outer circle is \(2r\) and, by Pythagoras' Theorem, the radius of the inner shaded circle is \(\sqrt{r^2 + r^2}\), that is \(\sqrt{2}r\).
So the radii of the two circles are in the ratio \(\sqrt{2} : 2\), that is \(1 : \sqrt{2}\), and hence the ratio of their areas is \(1 : 2\).
Solution © UK Mathematics Trust, from the IMC 2007 solutions.