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 and be the radii of the head and the tail, respectively.

Then

which we can write as

And

which we can write as

Solving this second equation we get , and substituting into the first equation gives .

Then area of the whole bird is equal to

where each of these are half or quarter circles, with the area of a whole circle being equal to . Then we can write the area of the whole bird as:

*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.
*

## Different answer

I love this! But I get a different answer (34pi). I get the radius of the wings to be 3 (you do too), hence area of 2 wings 9pi?

## Thanks for pointing this out…

Thanks for pointing this out - we forgot a factor of 1/2! We have fixed it now.

## Fun!

Fun!