Regular Polygon: worked example
| Number of sides (n) | 6 |
|---|---|
| Side length (s) | 5 |
| Area | 65 |
| Perimeter | 30 |
| Apothem | 4.3 |
| Interior angle (degrees) | 120 |
Regular Polygon: formula
A regular polygon has equally long sides; side count and side length determine its perimeter. A regular polygon has equal sides and equal angles, such as a hexagon, an octagon or a pentagon. The calculator finds the area, perimeter, apothem, the distance from the center to the midpoint of a side, and the interior angle.
The number of sides must be at least 3. Use one unit for the side length.
Regular Polygon: changing Number of sides (n)
| Number of sides (n) | Area | Perimeter | Apothem | Interior angle (degrees) |
|---|---|---|---|---|
| 4 | 25 | 20 | 2.5 | 90 |
| 6 (example) | 65 | 30 | 4.3 | 120 |
| 8 | 120.7 | 40 | 6 | 135 |
| 10 | 192.4 | 50 | 7.7 | 144 |
Regular Polygon: compare input ranges
| Input | Tested from | To | Area: low input | Area: high input | Swing |
|---|---|---|---|---|---|
| Number of sides (n) | 4 | 8 | 25 | 120.7 | 95.7 (147%) |
| Side length (s) | 4.5 | 5.5 | 52.6 | 78.6 | 26 (40%) |
Regular Polygon: individual input comparisons
| Input | Value | Area | Perimeter | Apothem | Interior angle (degrees) |
|---|---|---|---|---|---|
| Number of sides (n) | 4 | 25 | 20 | 2.5 | 90 |
| 6 | 65 | 30 | 4.3 | 120 | |
| 8 | 120.7 | 40 | 6 | 135 | |
| 10 | 192.4 | 50 | 7.7 | 144 | |
| Side length (s) | 4 | 41.6 | 24 | 3.5 | 120 |
| 4.5 | 52.6 | 27 | 3.9 | 120 | |
| 5 | 65 | 30 | 4.3 | 120 | |
| 5.5 | 78.6 | 33 | 4.8 | 120 | |
| 6 | 93.5 | 36 | 5.2 | 120 |
Regular Polygon: combined input changes
| Number of sides (n) / Side length (s) | 4 | 5 | 6 |
|---|---|---|---|
| 2 | Undefined for these inputs | Undefined for these inputs | Undefined for these inputs |
| 4 | 16 | 25 | 36 |
| 6 | 41.6 | 65 | 93.5 |
| 8 | 77.3 | 120.7 | 173.8 |
| 10 | 123.1 | 192.4 | 277 |
Regular Polygon: which assumptions move the result?
Regular Polygon: values and definitions
| Quantity and definition | Value used |
|---|---|
| Number of sides (n) — 3 or more. A hexagon has 6, an octagon has 8. | 6 |
| Side length (s) | 5 |
| Area | 65 |
| Perimeter | 30 |
| Apothem | 4.3 |
| Interior angle (degrees) | 120 |
Regular Polygon: half and double Number of sides (n)
| Number of sides (n) | Area |
|---|---|
| 3 | 10.8 |
| 6 | 65 |
| 12 | 279.9 |
Common error: Using the distance from the center to a corner in place of the side. That is the circumradius, which is larger than the side for most polygons and inflates the area.
The formulas apply only to regular polygons: irregular shapes must be divided into triangles. The apothem is not the same as the radius from the center to a vertex. Rounding matters with many sides.
Units are squared for the area.
Geometry calculators: Rhombus Area Calculator, Hemisphere Volume Calculator, Cone Frustum Calculator, Capsule Volume Calculator, Triangular Prism Volume Calculator, Law of Sines Calculator.
A regular polygon splits into identical triangles around its center, and each triangle's height is the apothem. Six sides of 5 units give a perimeter of 30, an apothem of about 4.33 and an area near 65. The interior angle follows from the side count alone: 120 degrees for a hexagon, 135 for an octagon.
As the sides multiply, the shape approaches a circle.
Confirm every side and every angle match. A shape with equal sides but different angles is a rhombus-like figure, and these formulas will not fit it.
See the circle area calculator for the limit, the equilateral triangle, rectangle and rhombus pages for specific polygons, and the arc length and sector calculator for circular pieces. The trigonometry calculator gives the angles, and the land acreage page changes units.
What is the interior angle of a regular hexagon?
120 degrees, since a hexagon's six interior angles add up to 720 degrees and are equal, and the general formula is n minus 2 times 180 over n.
What is an apothem?
The perpendicular distance from the center of a regular polygon to the midpoint of one of its sides.
How do I find the area of a regular polygon?
Multiply the perimeter by the apothem and divide by two, or use the formula with the number of sides and side length.