Here's a way to construct a decagon with ruler and compass. Recall that a decagon is a regular polygon with 10 sides.
- Draw a circle (c1) O center; chart diameter [IA] of this circle and the circle (c2) with center I and radius RN.
- Build a radius [IB] perpendicular to [IA]. Draw the right (RB). It cuts the small circle (c1) J and K with BJ <>
- Building the circle (c3) of center B through J. This circle meets the great circle (c2) in B1 and B9. Reporter distance BB1 on this great circle: there is this point B2. Build even B3, B4 etc..
Contrary to popular belief, all regular polygons are constructible with ruler and compass. For example, it is impossible to construct with ruler and compass a heptagon (7 sides regular polygon). The constructability or not a regular polygon with n sides is possible if the number n can be written as the product of a power (possibly zero) of 2 with one or more of Fermat numbers.
What do we Fermat numbers? It is simply a number that can be written as 2 ^ (2 ^ q) +1 (with q a natural number). For example, 5 is a Fermat number for 5 = 2 ^ (2 ^ 1) + 1. Returning to our
decagon. Why is he building it? Note that 10 = 2 x 5. It is the product of a power of 2 a Fermat number (5).
