Regular Polygon Calculator
Calculate a regular polygon's side length, perimeter, area, angles, circumradius, and apothem from one known measurement.
Polygon inputs
Calculated properties
For a regular nonagon with perimeter 18 inches, the area is 24.7273 square inches.
Property breakdown
| Property | Value | Formula from side length |
|---|---|---|
| Side length (a) | 2 in | a = P ÷ n |
| Perimeter (P) | 18 in | P = n × a |
| Area (A) | 24.7273 in² | A = n × a² ÷ [4 × tan(π ÷ n)] |
| Interior angle (α) | 140° | α = (n – 2) × 180° ÷ n |
| Exterior angle (β) | 40° | β = 360° ÷ n |
| Circumcircle radius (R) | 2.9238 in | R = a ÷ [2 × sin(π ÷ n)] |
| Incircle radius / apothem (r) | 2.7475 in | r = a ÷ [2 × tan(π ÷ n)] |
All values come from one canonical regular-polygon model. Length results use the selected unit; area uses the corresponding squared unit.
How to use this regular polygon calculator
What this calculator does
This calculator solves the main dimensions of a convex regular polygon: a closed two-dimensional figure whose sides have equal length and whose interior angles are equal. Give the number of sides and any one supported measurement, and the calculator derives the side length, perimeter, area, one interior angle, one exterior angle, circumcircle radius, and incircle radius or apothem. It is useful for geometry exercises, layout planning, fabrication estimates, pattern design, and checking hand calculations. It does not test whether a real-world object is perfectly regular, account for material thickness, or replace tolerances required for manufacturing or construction drawings. For the underlying definition and properties, see the Wolfram MathWorld regular polygon reference.
When to use it
Use the calculator when you know a regular polygon's perimeter but need its area; when a drawing specifies a circumradius or apothem and you need edge dimensions; when checking interior and exterior angles for a tile, sign, frame, or decorative panel; or when converting a polygon design between millimeters, centimeters, meters, inches, feet, and yards while preserving the same physical size.
How to calculate
- The calculator opens with a ready-to-use demonstration: a 9-sided regular polygon with an 18-inch perimeter. Its results are already calculated, and the example Excel workbook is immediately available.
- Replace Number of sides with a whole number from 3 through 10,000. A triangle is the smallest valid polygon.
- Choose the dimension you have under Known measurement, then enter it in Known value. Only one measurement is required because regular-polygon dimensions are linked by exact identities.
- Select the Length unit. A unit change converts the entered value rather than merely relabeling it. If the known measurement is area, the conversion uses squared units.
- Read the live result cards and the property table, then choose Download Excel to export the current typed inputs and canonical numeric results. Reset clears the demonstration and all calculated content; Download Excel then remains unavailable until a complete valid state is entered again.
Input guide
Number of sides is required and accepts only an integer from 3 to 10,000, such as 9. More sides increase each interior angle toward 180° and decrease each exterior angle toward 0°. Do not enter decimals, scientific notation, or a value below 3. Known measurement is required and selects the meaning of the numeric value: Side length, Perimeter, Area, Circumcircle radius, or Incircle radius / apothem. Choose the measurement actually supplied by your drawing; confusing radius with diameter is a common error. Known value is required, must be positive, and accepts U.S.-style decimal notation with optional correctly placed thousands separators, such as 18, 2.75, or 1,250.5. It rejects decimal commas, scientific notation, unit symbols, zero, and negative values. A larger known value scales all lengths proportionally and area by the square of the scale factor. Length unit is required and determines the displayed length and square-area units. For example, choose inches for an 18-inch perimeter. The metric conversion relationships are summarized by NIST's SI length guidance.
Output guide
Area is the enclosed surface measure in squared units and is the primary result. Side length is the length of each equal edge. Perimeter is the sum of all sides. Interior angle α is the angle inside the polygon at each vertex, while Exterior angle β is the turning angle between one side and the extension of the next; for a regular polygon, the two add to 180°. Circumcircle radius R runs from the center to a vertex. Incircle radius / apothem r runs from the center perpendicularly to the midpoint of a side. Every output is an exact geometric identity evaluated numerically, subject only to display rounding. OpenStax explains the perimeter and regular-polygon angle formulas and the apothem-based area formula.
Worked example
The first-open example uses 9 sides, selects Perimeter, enters 18, and uses inches. Dividing the perimeter by the side count gives a side length of 18 ÷ 9 = 2 in. The area formula gives 9 × 2² ÷ [4 × tan(π ÷ 9)] = 24.7273 in². The interior angle is (9 – 2) × 180° ÷ 9 = 140°, and the exterior angle is 360° ÷ 9 = 40°. The circumradius is 2 ÷ [2 × sin(π ÷ 9)] = 2.9238 in, while the apothem is 2 ÷ [2 × tan(π ÷ 9)] = 2.7475 in. These figures match the initial cards, table, and workbook checkpoints.
How the formulas connect
A regular polygon can be split into congruent isosceles triangles by drawing segments from its center to every vertex. Each central triangle has base a, height equal to the apothem r, and central angle 360° ÷ n. That construction explains why the area can be written both as one-half of perimeter times apothem and as a trigonometric expression in side length.
The circumradius reaches a vertex and is always at least as large as the apothem, which reaches a side. For any finite regular polygon with positive side length, both radii and the area must be positive. The calculator checks those invariants before it renders results or builds a workbook.
Interpretation and common mistakes
- Area changes quadratically. Doubling every length makes the area four times as large, not twice as large.
- Perimeter and side length are linear. Multiplying the side length by the number of sides gives the perimeter exactly.
- Do not use a diameter where a radius is requested. A diameter is twice the corresponding radius.
- Keep units consistent. A result in cm² is not directly interchangeable with m² without squaring the conversion factor.
- Very high side counts approximate a circle, but the result is still a polygon with straight edges.