What is the area of the bird, shown below?
Hint: There are two equal semicircles representing the wings, one semicircle representing the head, one quarter of circle representing the body (with a radius equal to the diameter of the head) and one quarter of circle representing the tail.
Solution
Let $R$ and $r$ be the radii of the head and the tail, respectively.
Then $$ \mbox{diameter of the left wing} + \mbox{diameter of the head} + \mbox{radius of the right wing} =17 $$ which we can write as $$ 2R-r+2R+\frac{2R-r}{2}=17. $$ And $$ \mbox{radius of the head} + \mbox{radius of the body} = 12 $$ which we can write as $$ R+2R=12. $$ Solving this second equation we get $R=4$, and substituting $R=4$ into the first equation gives $r=2$. Then area of the whole bird is equal to $$ \mbox{area of the head} + 2 \times \mbox{area of a wing} + \mbox{area of the body} + \mbox{area of the tail} $$ where each of these are half or quarter circles, with the area of a whole circle being equal to $\pi \times \mbox{radius}^2$. Then we can write the area of the whole bird as: $$ \begin{array}{ll} & \frac{\pi R^2}{2}+2\times\frac{\pi \left(\frac{(2R-r)}{2}\right)^2}{2}+\frac{\pi (2R)^2}{4}+\frac{\pi r^2}{4} \\ = & \frac{\pi 4^2}{2}+2\times\frac{\pi \left(\frac{(2(4)-2)}{2}\right)^2}{2}+\frac{\pi 8^2}{4}+\frac{\pi 2^2}{4} \\ = & 8\pi + 9\pi +16\pi +\pi \\ = & 34\pi. \end{array} $$
Thanks to Paulo Ferro for this puzzle - you can find it in his new geometry puzzle book: Birds, Bees and Burgers. You can find out more about it at his new website EnigMaths.