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Regular Polygon Calculator

Regular polygons appear across geometry, engineering, and design. This calculator determines all key properties—area, perimeter, interior and exterior angles, circumradius, and inradius—from just a single input value. Supply the number of sides and any one measurement (side length, perimeter, area, or radius), and the tool computes everything else. Essential for architects, students, and anyone working with geometric constructions.

Last updated: April 27, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

6,221 people find this calculator helpful

What is a Regular Polygon?

A regular polygon is a closed, two-dimensional shape bounded by straight line segments of equal length, with all interior angles identical. Unlike irregular polygons, where sides and angles vary, regular polygons possess perfect symmetry. Common examples include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), and hexagons (6 sides).

The defining properties of regularity make these shapes prevalent in nature, architecture, and mathematics. Snowflake crystals form hexagonal patterns, honeycomb cells are hexagonal for structural efficiency, and regular polygons tile planes without gaps. Each polygon can be inscribed within a circle (circumcircle) and circumscribed around a smaller circle (incircle), a relationship that proves crucial for calculating dimensions.

Polygons are named using Greek numerical prefixes combined with the suffix -gon. A heptagon has 7 sides, an octagon has 8, a nonagon has 9, and a decagon has 10. Understanding polygon classification helps identify the number of sides and predict angular properties before calculation.

Regular Polygon Formulas

All properties of a regular polygon depend on two variables: the number of sides (n) and any single measurable parameter. Below are the fundamental equations governing perimeter, area, and angular relationships.

Perimeter = n × a

Area = (n × a² × cot(π/n)) ÷ 4

Interior angle (α) = ((n − 2) × π) ÷ n

Exterior angle (β) = 2π ÷ n

Circumradius (R) = a ÷ (2 × sin(π/n))

Inradius (r) = a ÷ (2 × tan(π/n))

n — Number of sides of the polygon
a — Length of one side
π — Mathematical constant (approximately 3.14159)
α — Interior angle (angle between adjacent sides inside the polygon)
β — Exterior angle (supplementary to interior angle)
R — Circumradius—radius of the circumscribed circle passing through all vertices
r — Inradius (apothem)—radius of the inscribed circle tangent to all sides

Polygon Names and Side Counts

Identifying a polygon by name reveals its side count. Below are common regular polygons encountered in mathematics and design:

Triangle (3 sides): Interior angle 60°, exterior angle 120°
Square (4 sides): Interior angle 90°, exterior angle 90°
Pentagon (5 sides): Interior angle 108°, exterior angle 72°
Hexagon (6 sides): Interior angle 120°, exterior angle 60°
Heptagon (7 sides): Interior angle 128.57°, exterior angle 51.43°
Octagon (8 sides): Interior angle 135°, exterior angle 45°
Nonagon (9 sides): Interior angle 140°, exterior angle 40°
Decagon (10 sides): Interior angle 144°, exterior angle 36°

As the number of sides increases, interior angles approach 180° and the polygon becomes increasingly similar to a circle.

How to Use the Calculator

The calculator requires two inputs: the number of sides and one additional parameter. Follow these steps:

Enter the number of sides. Type the value for n (e.g., 6 for a hexagon).
Provide one known measurement. Input any single value: side length, perimeter, area, circumradius, inradius, or an angle.
Calculate instantly. The tool derives all remaining properties automatically using the geometric relationships above.

Example: For a nonagon (9 sides) with a perimeter of 18 cm, the calculator determines: side length = 2 cm, area ≈ 24.73 cm², interior angle ≈ 140°, exterior angle = 40°, circumradius ≈ 2.92 cm, and inradius ≈ 2.75 cm.

Key Considerations

Avoid common mistakes when working with regular polygons.

Angles are measured in radians in formulas — Mathematical formulas use radians (where π radians = 180°), not degrees. When calculating interior and exterior angles, convert between radians and degrees as needed. For a hexagon, the interior angle is 2π/3 radians, equivalent to 120°.
Apothem and inradius are identical — The inradius (the radius of the inscribed circle) is also called the apothem—the perpendicular distance from the polygon's center to the midpoint of any side. Both terms describe the same measurement.
One input is sufficient — Unlike some calculators, you only need one parameter (side, perimeter, area, or radius) plus the side count. Providing conflicting values for two parameters will cause errors; verify your input matches the polygon you're analyzing.
Exterior angles always sum to 360° — The sum of all exterior angles in any polygon is always 360°. For a regular polygon with <em>n</em> sides, each exterior angle equals 360°/<em>n</em>. This relationship is independent of side length and useful for checking calculations.

Frequently Asked Questions

What's the difference between circumradius and inradius?

The circumradius (R) is the distance from the polygon's center to any vertex, defining the circle that passes through all corners. The inradius (r), also called the apothem, is the perpendicular distance from the center to the midpoint of any side, defining the inscribed circle that touches all edges. For a regular polygon, R is always larger than r, and their ratio depends on the number of sides.

How do I calculate the area of a regular polygon if I only know the perimeter?

First, derive the side length by dividing perimeter by the number of sides: side = perimeter ÷ n. Then apply the area formula: area = (n × side² × cot(π/n)) ÷ 4. For example, a regular pentagon with perimeter 50 cm has side length 10 cm, yielding area ≈ 172.05 cm². The relationship is direct because perimeter and side are proportional.

Why do interior angles increase as a polygon gains more sides?

Interior angles approach 180° as the number of sides increases toward infinity. This occurs because the polygon's shape becomes increasingly similar to a circle. A triangle's interior angles sum to 180°, a quadrilateral's to 360°, and the total is always (n − 2) × 180°. Distributing this total across more sides means each angle grows larger, but never exceeds 180° for a convex polygon.

Can irregular polygons use these formulas?

No. These formulas apply exclusively to regular polygons where all sides are equal and all angles are identical. Irregular polygons require different approaches: you might divide the shape into triangles and sum their areas, or use coordinate geometry. This calculator is designed for perfect regularity; irregular shapes demand custom analysis.

What does it mean if I get a non-integer number of sides?

It doesn't make physical sense—polygons must have whole numbers of sides. If your calculation yields a fractional result (e.g., 5.3 sides), it indicates an inconsistency in your input data. Verify that your provided measurement (area, perimeter, or radius) actually corresponds to a regular polygon with an integer side count.

How does the calculator handle very large polygons?

As the number of sides increases (30, 50, 100+), the regular polygon increasingly approximates a circle. The circumradius and inradius converge toward the same value, and the area calculation becomes nearly indistinguishable from circle area (πr²). The formulas remain mathematically valid for any positive integer, but practical precision depends on computational rounding at extreme values.

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