Regular polygon calculator
Area, perimeter, apothem and angles of a regular polygon.
Fill in the fields and the result will appear here automatically.
Works a regular polygon — equal sides and equal angles: a hexagonal tile, an octagonal gazebo, a triangular or pentagonal plot. The number of sides must be a whole number and at least three: two segments cannot enclose a polygon, and a fractional side count has no meaning. The interior angle is reported in degrees even though the area uses a tangent in radians — the two measures must never be mixed up.
How it works
Formula and logic
S = n · a² ÷ (4 · tan(π ÷ n)), P = n · a and the apothem is m = a ÷ (2 · tan(π ÷ n)); the interior angle is (n − 2) · 180° ÷ n.
Example
A regular hexagon with a side of 2 cm has an area of 10.392 cm² and an interior angle of 120°.
Fields and units
- Length unit — list option
- Number of sides — 1
- Side length — cm
How to use
- — Choose the length unit.
- — Enter the number of sides — a whole number, at least three.
- — Enter the side length and read the area.
- — This page accepts 3–1000 sides and rejects fractions. All sides and interior angles must be equal. Changing the length unit does not convert the entered side.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: sine and tangent in a right triangle
- Limitation
- A simple convex regular polygon with positive side length; star polygons and irregular outlines are not calculated. The limit n ≤ 1000 belongs to this page. Area uses the selected unit squared and angles use degrees.
FAQ
Why can I not enter two sides?
Two segments cannot enclose a figure: a polygon starts at three sides, and that is a definition rather than a limit of the calculator.
What is the apothem?
The apothem m is the centre-to-side-midpoint distance, the inradius. Area is Pm/2. To fit the whole polygon into a concentric circular opening, use circumradius a/[2sin(π/n)], which is larger than m.
Why must the side count be a whole number?
A side either exists or it does not; half a side is meaningless for a polygon, so a fractional value is rejected.
What unit is the angle in?
Degrees. Internally the area uses a tangent of radians, but the reported angle is converted to the familiar degrees.