Regular Polygon Formulas (Area, Perimeter, Angles)

A regular polygon has n equal sides of length s and n equal angles. Its perimeter is P = n·s and its area is A = ½·a·P, where a = s / (2·tan(180°/n)) is the apothem. Each interior angle equals (n−2)·180°/n.

Regular Polygon formulas

Perimeter: $P = n s$
Apothem: $a = \dfrac{s}{2\tan(180^\circ/n)}$
Area: $A = \tfrac{1}{2} a P$
Interior angle: $\dfrac{(n-2)\cdot 180^\circ}{n}$

where n = number of sides, s = side length, a = apothem (center to edge midpoint), P = perimeter

Perimeter

P = n s

Apothem

a = \dfrac{s}{2\tan(180^\circ/n)}

Area

A = \tfrac{1}{2} a P

Interior angle

\dfrac{(n-2)\cdot 180^\circ}{n}

Worked example

A regular hexagon (n = 6) has interior angles of 120° each and perimeter 24.

\dfrac{(6-2)\cdot 180^\circ}{6} = \dfrac{720^\circ}{6} = 120^\circ

Interactive regular polygon explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Number of sides n

Side length s

Perimeter P 24units

Apothem a 3.46units

Area A 41.57units²

Interior angle ∠ 120°

Step-by-step substitution

P = n · s = 6 · 4 = 24
a = s / (2 tan(180°/n)) = 3.46
A = ½ · a · P = ½ · 3.46 · 24 = 41.57
interior ∠ = (n−2)·180°/n = 120°

Classroom use: regular polygon formulas

Who it's for: Grades 8–12 · Students investigating interior angle sums and, in high school, the apothem area formula.

Learning objectives

After using this page, students will be able to:

Compute the interior angle of a regular n-gon.
Find area from the apothem and perimeter using A = ½ap.
Describe how a regular polygon approaches a circle as n increases.

Suggested classroom activity

Step n from 3 upward and have students tabulate the interior angle and area at a fixed radius. Watching the area creep toward πr² is the informal limiting argument for the circle formula — the best available preview of calculus ideas at this level.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`8.G.A.5`	Use informal arguments to establish facts about angle sums and exterior angles of polygons.
`HSG.MG.A.1`	Use geometric shapes, their measures, and their properties to describe objects.
`HSG.GMD.A.1`	Give an informal argument for circumference, area, and volume formulas.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

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Frequently asked questions

What is the interior angle of a regular polygon?

Each interior angle of a regular n-sided polygon is (n−2)·180°/n. A pentagon (n = 5) has angles of 108°; a hexagon (n = 6) has angles of 120°.

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