We shall first study regular polygons. A regular polygon is a polygon in which all sides have the same length and all angles are equal in measure. A rhombus is not a regular polygon, though all sides are equal. This is because all angles are not equal. Let be the side length of an -sided polygon (-gon), the circumradius, and the inradius.
An -gon is made up of isosceles triangles with base sides and angle between the two sides measuring
For relations between various lengths and angles, the right-angled triangle, which is half of the isosceles triangle, is helpful. So we get the relation and The base angles are while the angle between adjacent sides is
The sketch below shows how the isosceles triangles are placed with a common vertex, coinciding sides, and the angles. Apart from the solution of the Isosceles triangle, the bases of -gon solution, the lengths of diagonals are important.
For an -gon, there will be vertices Then lines can be drawn from to othere vertices. Out of these, and are adjacent sides. So there are only diagonals. If is even, there will be one big diagonal, and all the remaining ones will be in pairs, as shown in the sketch. If is odd, all diagonals will be in pairs. What is true for vertex is true for each of the vertices.
Next, we have to find out the lengths of diagonals.
The diagonal between also forms an isosceles triangle with sides and the angle Now we can solve these as we had solved for sides.