Oh so circular
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?
Age
14 to 16
Challenge level
Problem
Image
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?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Image
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 $1:2$.
Since the area of the inner circle is $1$, the area of the outer circle is $2$.