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How big is the bird? – solution

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How big is the bird? – solution

What is the area of the bird, shown below?

a little bird

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.

a little bird

Then diameter of the left wing+diameter of the head+radius of the right wing=17 which we can write as 2Rr+2R+2Rr2=17. And radius of the head+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 area of the head+2×area of a wing+area of the body+area of the tail where each of these are half or quarter circles, with the area of a whole circle being equal to π×radius2. Then we can write the area of the whole bird as: πR22+2×π((2Rr)2)22+π(2R)24+πr24=π422+2×π((2(4)2)2)22+π824+π224=8π+9π+16π+π=34π.

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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.

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Comment

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?