The Length of a Side of a Regular Polygon calculator computes the length of the individual sides (segments) of a regular polygon given the number of sides and the radius, r, of an inscribed circle.
INSTRUCTIONS: Choose units and enter the following:
- (r) This is the inner radius
- (n) This is the number of sides of the polygon
Polygon Side Length (s): The calculator returns the length of the side in meters. However, this can be automatically converted to compatible units via the pull-down menu.
Polygon Calculators:
- Area of a Polygon based on the length and number of sides.
- Area of a Polygon based on the number of sides and the outer radius.
- Area of a Polygon based on the number of sides and the inner radius.
- Perimeter of a Polygon based on the number and length of sides.
- Perimeter of a Polygon based on the number of sides and the outer radius.
- Perimeter of a Polygon based on the number of sides and the inner radius.
- Length of a Polygon Side based on Circumscribed Circle Radius
- Length of a Polygon Side based on Inscribed Circle Radius
The Math / Science
A regular n-sided polygon is a polygon with n equal length sides and has n equal angles at the n vertices of the polygon. Because of the symmetries of this construction, the sides of the polygon touch the inscribed circle as tangents at the midpoint of the polygon side. As shown in the picture, Figure 1, lines from the vertices to the circle's center form n isosceles triangles of equal area.
The formula for the length of a side of a polygon based on the inner radius and number of sides is:
where:
- s is the length of the sides of a polygon outside of the circle
- r is the radius of the inner circle
- n is the number of sides (integer)
Derivation
The n-sided 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 the circle's radius. It also can be shown that the radius splits each sector of the circle in half, and thus the triangles composing the polygon are also split into two equal right triangles. 
If we imagine the sides of the polygon as the bases of these isosceles triangles, then we can see the area of each of the n triangles is given by the simple formula:
[1] `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:
[2] `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:
[3] `tan(alpha/2) = (1/2s) / r`
Substituting equation [2] into equation [3] we get:
[4] `tan( ((2*pi) / n)/2 ) = s / (2*r)`
And rearranging we finally get the polygon side, s, in terms of the radius, r, and the number of sides of the regular polygon, n:
[5] `s = 2*r *tan( pi / n)`