icon

How big is the bird? – solution

Share this page

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

  \[  \mbox{diameter of the left wing} + \mbox{diameter of the head} + \mbox{radius of the right wing} =17  \]    

which we can write as

  \[  2R-r+2R+\frac{2R-r}{2}=17.  \]    

And

  \[  \mbox{radius of the head} + \mbox{radius of the body} = 12  \]    

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

  \[  \mbox{area of the head} + 2 \times \mbox{area of a wing} + \mbox{area of the body} + \mbox{area of the tail}  \]    

where each of these are half or quarter circles, with the area of a whole circle being equal to $\pi \times \mbox{radius}^2$. 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.

Permalink
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?

Permalink
Comment

Fun!

  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.

  • The BloodCounts! project is gearing up towards one of the largest-scale applications yet of machine learning in medicine and healthcare.

  • What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.

  • Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!

  • How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?

  • Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.