Regular polygon angles, apothem and area
How side count sets the interior and exterior angles of a regular polygon, and how apothem and circumradius build the area formula.
Interior and exterior angles
Every regular polygon splits its total angle budget evenly among its corners. The interior angles of an n sided polygon sum to (n minus 2) times 180 degrees, so each one equals that total divided by n. The exterior angles are simpler still: they always sum to exactly 360 degrees no matter how many sides there are, so each exterior angle is 360 over n. Since an interior and its exterior angle sit on a straight line, they add to 180 degrees. That is why a triangle has 60 degree interiors, a square has 90, and a decagon has 144.
The apothem and the tangent
The apothem is the perpendicular distance from the centre to the middle of a side, and it is the key to the area. Drop a line from the centre to a side midpoint and another to a nearby vertex, and you form a right triangle whose central angle is pi over n radians, or 180 over n degrees. Half the side is the opposite leg and the apothem is the adjacent leg, so the apothem equals the side divided by twice the tangent of pi over n. As the side count grows the apothem approaches the circumradius, because the flat edges hug the circle ever more tightly.
Why area is half perimeter times apothem
Cut a regular polygon into n triangles by joining the centre to every vertex. Each triangle has a base equal to the side length and a height equal to the apothem, so its area is one half base times height. Adding all n triangles multiplies that by n, and n times the side is just the perimeter. The result collapses to one half times perimeter times apothem, a formula that works for any regular polygon. For the pentagon of side 6 the perimeter is 30 and the apothem is about 4.129146, giving an area near 61.937186.
Circumradius and the two radii
The circumradius reaches from the centre to a vertex and equals the side divided by twice the sine of pi over n. It is the radius of the circle that passes through every corner, while the apothem is the radius of the circle that just touches every edge. The two are linked by the cosine of pi over n, since apothem equals circumradius times that cosine. In the hexagon this cosine is about 0.866, and the circumradius happens to equal the side exactly, a coincidence unique to six sides among the common polygons.