The Area of Polygon Segment from Number of Sides and Inner Radius calculator computes the area of one of n triangular segments of a regular polygon of n sides circumscribed around a circle of radius r. See the colored segment of the polygon shown in Figure 1.
INSTRUCTIONS: Choose units and enter the following:
Polygon Segment Area (A): The calculator returns the area in square meters. However, this can be automatically converted to compatible units via the pull-down menu.
A regular n-sided polygon is a polygon with n equal length sides and is a polygon which has n equal angles at the n vertices of the polygon. Because of the symmetries of this construction, shown in Figure 1, the sides of the polygon touch the inscribed circle as tangents at the midpoint of the polygon side. As shown in the Figure 1, lines from the vertices of the regular polygon to the circle's center form n isosceles triangles of equal area. The segment highlighted in Figure 1 is one of these isosceles triangles and this equation computes this triangle's area.
The n-sided area of a regular polygon, as can be seen in Figure 1, is comprised of n isosceles triangles. The radius of the circle is constructed to touch the side of the polygon at exactly one point, and thus the polygon's side touches the circle at a tangent point and this makes the side of the polygon perpendicular to a radius. The lines from the circle center to the polygon vertices define the sides of n equal sectors of the circle. It also can be shown that a radius to the tangent point on the polygon splits each sector of the circle into two equal sectors. The n isosceles triangles composing the polygon are thus also split into two equal right triangles by the lines between the circle center and the polygon vertices.
If we imagine a side of the polygon as the base of one of these isosceles triangles (e.g., the triangle labeled Segment) ,then we can see the area of the triangles is given by the simple formula:
 `A_"(triangle)" = 1/2 * base * height = s/2 * r`
We can find the length of the polygons side s by noting first that the triangle with base s/2 and height r is a right triangle. We also note that the angle, `alpha`, is given by:
 `alpha = (2 * pi) /n`
We also see that line L is the hypotenuse of the right triangle and thus is related to r and s/2 as:
 `tan(alpha/2) = (1/2s) / r`
Substituting equation  into equation  we get:
 `tan( ((2*pi) / n)/2 ) = s / (2*r)`
And rearranging we get the polygon side, s, in terms of the radius, r, and the number of sides of the regular polygon, n:
 `s = 2*r *tan( pi / n)`
Finally to get area, we substitute equation  into equation :
 `A = 1/2* (2*r *tan( pi / n)) * r`
And simplifying, we get the area of each one of the n isosceles triangles comprising the regular polygon:
 `A_"(triangle)" = r^2 tan( pi / n)`