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 $$\text{diameter of the left wing}+\text{diameter of the head}+\text{radius of the right wing}=17$$ which we can write as $$2R-r+2R+\frac{2R-r}{2}=17.$$ And $$\text{radius of the head}+\text{radius of the body}=12$$ 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 $$\text{area of the head}+2\times \text{area of a wing}+\text{area of the body}+\text{area of the tail}$$ where each of these are half or quarter circles, with the area of a whole circle being equal to $\pi \times {\text{radius}}^2$. Then we can write the area of the whole bird as: $$\begin{array}{ll}& \frac{\pi R^2}{2}+2\times \frac{\pi {\left(\frac{(2R-r)}{2}\right)}^2}{2}+\frac{\pi (2R{)}^2}{4}+\frac{\pi r^2}{4}\\ =& \frac{\pi 4^2}{2}+2\times \frac{\pi {\left(\frac{(2(4)-2)}{2}\right)}^2}{2}+\frac{\pi 8^2}{4}+\frac{\pi 2^2}{4}\\ =& 8\pi +9\pi +16\pi +\pi \\ =& 34\pi .\end{array}$$

Back to original puzzle

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.