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
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which we can write as
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And
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which we can write as
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Solving this second equation we get , and substituting
into the first equation gives
.
Then area of the whole bird is equal to
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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:
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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!