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`s = 2 * r * sin(pi/ n )`

Enter a value for all fields

The **Length of the Sides of a Polygon based on the Outer Radius **calculator computes the length of the individual sides (segments) of a regular polygon given the number of sides of the regular polygon and the radius, **r**, of a circumscribed (outer) circle.

**INSTRUCTIONS**: Choose units and enter the following:

- (
**r**) This is the outer radius - (
**n**) This is the number of sides of the polygon

**Polygon Side Length (s): **The calculator returns the length of the sides in meters. However, this can be automatically converted to compatible units via the pull-down menu.

- 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

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 symmetry of this construction, all the vertices of the regular polygon lie on the circle and the sides of the regular polygons form **n** chords of the circle.

The formula for the length of a side of a polygon based on the outer radius and number of sides is:

where:

- s is the length of the sides of a polygon inside of the circle
- r is the radius of the outer circle
- n is the number of sides (integer)

The n-sided area of a regular polygon, as can be seen in Figure 1, is comprised of **n** isosceles triangles. The regular polygon is constructed to have all its vertices on the circle, and thus a radii of the circle intersects a vertex of the polygon and bisects the angle of the polygon. It also can be shown that the radii to the polygon's vertices form the sides of **n** isosceles triangles whose third side is the polygon's side. The line from the center to the s/2 point on the polygon side splits these triangles into two right triangles.

If we imagine the sides of the polygon as the bases of these isosceles triangles, we can find the length of the polygon's side **s** by noting first that the triangle with base **s/2** and height **L **is a right triangle. We also note that the angle, `alpha`, is given by:

[2] `alpha = (2 * pi) /n`, because all the **n** equal angles `alpha` must sum to `2pi` radians

We then see that the circle's radius, **r**, is the hypotenuse of a right triangle and thus relates** r **and **s/2** as:

[3] `s/2 = r * sin (alpha/2)`

Substituting equation [2] into equation [3] we get:

[4] `s/2 = r *sin( ((2*pi) / n)/2 )`

And rearranging we get the polygon side, **s**, in terms of the circle's radius, **r**, and the number of sides of the regular polygon, **n**:

[5] `s = 2 * r * sin( pi / n)`