How big is the bird? – solution
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.