A regular hexagon
For n = 6 and side s = 4, perimeter is 24, circumradius is 4, and apothem is 2√3. Area is 24√3, approximately 41.56921938 square units.
Regular Polygon Calculator inputs
Changing the unit relabels the numbers. Convert measurements to one common unit before entering them.
Use 3–1,000 equal sides. Circumradius reaches a vertex; the apothem meets a side at 90°. These two center-to-edge measurements are different. Positive measurements must be from 0.000000001 to 1,000,000,000; derived defining lengths must also remain in this range.
Calculated measurements
Area
41.56921938 units²
| Measurement | Value |
|---|---|
| Area | 41.56921938 units² |
| Perimeter | 24 units |
| Side length | 4 units |
| Circumradius (center to vertex) | 4 units |
| Apothem (center to side) | 3.464101615 units |
| Interior angle | 120.000000° |
| Central angle | 60.0000000° |
| Number of diagonals | 9 |
Formula: P = ns; apothem = s/[2 tan(π/n)]; A = P × apothem/2
All sides and all interior angles are equal. The diagram draws up to 80 vertices; larger polygons are represented by a labeled circle.
Results use full floating-point precision internally and display up to 10 significant digits. Angles are in degrees.
Calculations run locally in your browser. Measurement values are not sent to a calculation service. The downloadable text contains your selected inputs and computed results.
How to use the Regular Polygon Calculator
- 01
Enter a whole number of sides from 3 through 1,000.
- 02
Choose side length, circumradius, or apothem as the known size.
- 03
Enter a positive length in a common unit and inspect the labeled polygon.
- 04
Read area, perimeter, angles, and diagonal count, then copy or download the results.
Examples
A square from its side
For n = 4 and s = 5, area is 25 and perimeter is 20. The apothem is 2.5 and circumradius is 5/√2, approximately 3.535533906.
An equilateral triangle
For n = 3 and s = 6, each interior angle is 60° and the diagonal count is zero. Its area is 9√3, approximately 15.58845727 square units.
Regularity is a stronger condition
Equal sides alone do not guarantee a regular polygon; all interior angles must also be equal. A rhombus is a useful counterexample when its angles are not right angles. The calculator uses the side count and one size to construct a regular shape rather than testing measured vertices for regularity.
Apothem and circumradius
Draw a segment from the center to a vertex for circumradius R. Draw the perpendicular from the center to a side for apothem a. Splitting one central sector gives a right triangle with angle π/n, so s = 2R sin(π/n) and a = s/[2 tan(π/n)]. The apothem is smaller than the circumradius.
Area follows from equal triangles
A regular n-gon divides into n congruent triangles, each with base s and height a. Their total area is nsa/2, or perimeter times apothem divided by two. This approach also explains why multiplying a polygon’s perimeter by its circumradius would produce the wrong area.
Angles, diagonals, and the drawing limit
The central angle is 360°/n and each interior angle is 180° − 360°/n. A diagonal connects nonadjacent vertices, giving n(n − 3)/2 diagonals. The tool calculates up to 1,000 sides but draws individual vertices only up to 80 sides; above that, a labeled circle approximation keeps the interface responsive.
What to know
- Convex regular polygons only; star polygons and irregular figures are excluded.
- Side, apothem, and circumradius must each fit 10⁻⁹–10⁹ units. The drawing is approximate above 80 sides.
Common questions
Can this solve an irregular polygon?
No. An irregular polygon generally needs vertex coordinates or additional measurements. One side length cannot establish its area.
Is the apothem the same as the radius?
No. Apothem reaches the midpoint of a side at a right angle. Circumradius reaches a vertex. They approach each other as the side count grows.
Why is the side count limited?
The 3–1,000 range keeps the supported geometry and display bounded. The diagram switches to a circle approximation above 80 sides.
How many diagonals does a pentagon have?
Five. The formula n(n − 3)/2 gives 5 × 2/2. Sides and center-to-vertex segments are not counted as polygon diagonals.