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How to use
Every side and angle of a regular polygon is equal. The side count plus one known size — side, apothem, circumradius or area — is enough to recover the rest, including angles and diagonal count.
- Enter an integer side count from 3 to 10,000.
- Choose the known measurement and enter a positive value.
- Choose the underlying length unit.
- Review area, perimeter, apothem, radius, angles and diagonal count.
How it is calculated
Example
For a regular hexagon with side 10: perimeter 60, interior angle 120°, exterior angle 60°, apothem ≈8.660, circumradius 10 and area ≈259.808.
Important notes
These formulas are for convex regular polygons, not star polygons or irregular shapes. If area is the known input, it is interpreted in the square of the selected length unit.
Worked examples and interpreting results
A regular hexagon with side length 2 has perimeter 12 and area 6√3. Regularity means equal sides and equal angles, allowing side count and one side length to determine the shape.
| Case | Calculation | Result |
|---|---|---|
| Perimeter | 6 × 2 | 12 |
| Apothem | 2 ÷ (2 × tan(π/6)) | 1.732051 |
| Area | 12 × √3 ÷ 2 | 10.392305 |
How to check the result
Do not use this formula for equal-sided polygons with unequal angles. The apothem is perpendicular to a side and differs from the radius to a vertex. Side count must be an integer of at least three.
Frequently asked questions
What is the apothem?
The perpendicular distance from the center to a side midpoint; it is the inscribed-circle radius.
How many diagonals are there?
n(n−3)/2, counting non-adjacent vertex connections once.
What is the difference between circumradius and apothem?
The circumradius runs from the center to a vertex, while the apothem runs perpendicularly from the center to the midpoint of a side. They approach each other as side count grows but are not generally equal.