Regular polygon calculator

Calculate the area, perimeter, angles, circumradius, apothem, and side length of a regular polygon with live geometry and clear visual explanations.

⚑ Report error
Area
—
n·a²·cot(π/n) / 4
Perimeter
—
n × side
Interior angle
—
(n − 2)·180° / n

Live regular polygon

Vertices, circumcircle, apothem and side update with n
Live
n · a · R · rUpdates with every input

Polygon inside its circumcircle

Polygon area compared with the enclosing circle
Live
polygon area + remaining circle—

Angles as the number of sides changes

Interior and exterior angles across regular polygons
Live
α = interior · β = exteriorCurrent n marked
LIVE GEOMETRY CHECK

What the polygon is telling you

Updates live
Apothem / radius—r ÷ R
Central angle—360° ÷ n
Circle coverage—polygon area ÷ πR²
Triangle pieces—n isosceles triangles
Change n or one known length to see the geometry respond.

Regular polygon properties

n = 6
PropertySymbolFormulaResult

Polygon parameters

Use any whole number from 3 to 1000.
cm
cm
cm²
cm
cm
deg
deg
Enter n and any one known length or property. The calculator derives the remaining regular-polygon properties.
THE CONCEPT

What is a regular polygon?

A regular polygon is a closed plane shape with the same length on every side and the same measure for every interior angle. Equilateral triangles, squares, regular pentagons, hexagons, octagons and many other n-gons are regular polygons.

The key to understanding a regular polygon is its center. Draw segments from the center to every vertex and the polygon breaks into n congruent isosceles triangles. That one construction connects the side length, circumradius, apothem, area, perimeter and angles.

R r a center One polygon → n equal triangles central angle = 360° / n area = n × (a × r) / 2 perimeter = n × a R = a / [2 sin(π/n)] r = a / [2 tan(π/n)]
For a regular polygon, the center creates a repeatable triangle pattern. This is why one side length plus n is enough to reconstruct the entire shape.
NAMES & STRUCTURE

From triangle to n-gon

The number of sides determines the familiar name and the characteristic angles. The same formulas work for every regular n-gon; only n changes.

PolygonnInterior αExterior βCentral angle
Equilateral triangle360°120°120°
Square490°90°90°
Pentagon5108°72°72°
Hexagon6120°60°60°
Heptagon7128.57°51.43°51.43°
Octagon8135°45°45°
Nonagon9140°40°40°
Decagon10144°36°36°
Dodecagon12150°30°30°
FORMULAS

Regular polygon formulas

Let n be the number of sides, a the side length, R the circumradius and r the inradius (apothem).

Perimeter

P = n a

Add n equal sides.

Area from side

A = n a² cot(π/n) / 4

Useful when side length is known.

Area from apothem

A = P r / 2

Split the polygon into n triangles.

Inradius / apothem

r = a / [2 tan(π/n)]

Center-to-side distance.

Circumradius

R = a / [2 sin(π/n)]

Center-to-vertex distance.

Angles

α = (n−2)180°/n
β = 360°/n

Interior and exterior angles.

THE GEOMETRY

Apothem, circumradius and the side

Three lengths appear repeatedly in regular-polygon geometry. The circumradius R reaches from the center to a vertex. The apothem r reaches perpendicularly from the center to the midpoint of a side. The side length a connects adjacent vertices.

R r a R and r are not the same radius R → center to a vertex r → center to a side midpoint a → one side of the polygon r = R cos(π/n)
Dropping a perpendicular from the center to a side creates the apothem. Together with half a side, it forms a right triangle that explains the sine and tangent formulas.
ANGLES

Interior, exterior and central angles

The interior angles become larger as n increases, while each exterior and central angle becomes smaller. Three different angles are useful, and confusing them is one of the most common regular-polygon mistakes.

Interior α(n−2)180°/ninside each vertex
Exterior β360°/none turn around the polygon
Central θ360°/nbetween adjacent radii
θ exterior β interior α A useful identity α + β = 180° β = θ = 360°/n Sum of interior angles = (n−2)180°
EXAMPLES

Worked examples

Example 01 · Hexagon

Side = 10 cm, n = 6

P = 6×10 = 60 cm.

A = 6×10²×cot(30°)/4 ≈ 259.81 cm².

Interior angle = 120°, exterior angle = 60°.

Example 02 · Pentagon

Perimeter = 50 cm, n = 5

Side a = P/n = 50/5 = 10 cm.

Then r and R follow from the tangent and sine formulas.

Example 03 · Octagon

Side = 4 m, n = 8

P = 32 m.

Interior angle = 135° and exterior angle = 45°.

Area ≈ 77.25 m².

REAL-WORLD USE

Where regular polygons are used

Regular polygons are not only classroom shapes. Their symmetry makes them useful in signs, decorative tiling, product design, architecture and technical drawings.

Real octagonal stop sign
Octagon · road signsAn octagon is immediately recognizable as a stop-sign silhouette.
Octagonal floor tile pattern
Octagons · flooringRegular octagons can repeat with small square infill tiles.
SELECTED POLYGONS

A quick reference table

NamenInterior αExterior βCommon example
Triangle360°120°Equilateral triangle
Square490°90°Tile / frame
Pentagon5108°72°Symmetric motifs
Hexagon6120°60°Honeycomb-inspired design
Octagon8135°45°Stop sign
Decagon10144°36°Decorative geometry
Dodecagon12150°30°Architectural details
CHECK YOUR WORK

Common mistakes

1. Mixing α and β

Interior and exterior angles are supplementary at a vertex: α + β = 180°.

2. Confusing R and r

R reaches a vertex; r reaches a side midpoint.

3. Forgetting the square

When n = 4, the regular polygon is a square and the general formulas still work.

4. Mixing units

Use the same length unit for side, radius and perimeter. Area uses the squared unit.

FAQ

Regular polygon calculator FAQ

What makes a polygon regular?

Every side has the same length and every interior angle has the same measure.

What is an apothem?

The apothem is the perpendicular distance from the center of a regular polygon to one of its sides. It is also the inradius of the inscribed circle.

Can I calculate the polygon from perimeter instead of side length?

Yes. With n known, the side is simply a = P/n. The remaining properties can then be derived.

What happens as n becomes very large?

The polygon approaches a circle when its circumradius is held fixed. The live circle-coverage chart makes this convergence visible.

What is the difference between circumradius and inradius?

The circumradius passes through every vertex; the inradius (apothem) touches every side at its midpoint.