Archive for March, 2013

happy “Pi” day

Posted: March 14, 2013 in Mathematics

dodecagonToday I am going to show you how to find the exact area of a regular dodecagon with side length of 1 without using a calculator.  First of all, let me tell you what a regular dodecagon is.  A regular dodecagon is a regular 12-gon, a 2D shape with 12 equal sides.

Let’s begin!

See the diagram, a regular dodecagon consists of 12 isosceles triangles with central angle of 30 degrees.  Let the base of that triangle be 1, then




then the area of one triangle is h/2 and the exact area of the regular dodecagon is 6h.  Now, the difficult part is to find the exact value of h by hand.  (Remember this is a math contest question, no calculator is allowed.  Just your brain, a pencil, and papers. )

Now, if it is tan(30), it will be easy because \tan{30^\circ}=\dfrac{1}{\sqrt{3}}

but notice that 15=30/2, so we can use the half angle trig identity.





then we have




And since our angle here is less than 90 degrees, we can drop the negative sign when taking square roots.



then replace x with x/2 yields,



Then tangent is sine over cosine,


Then substitute in x=30, we have






Therefore, the area of the regular dodecagon with side length of 1 is