What is a regular polygon?

A regular polygon is a polygon where all sides are equal in length and all interior angles are equal. Examples include equilateral triangles, squares, and regular hexagons.

What is the formula for the area of a regular polygon?

If n is the number of sides and s is the side length, the area of a regular polygon is A = n·s² / (4·tan(π/n)). If you know the apothem a and perimeter P, you can also use A = P·a / 2.

How do I find the area of a regular polygon from the radius?

If R is the circumradius (distance from center to a vertex) and n is the number of sides, the area is A = n·R²·sin(2π/n) / 2. The calculator can do this automatically when you choose the radius input mode.

What units does the regular polygon area use?

The area is always in squared units of whatever length unit you use for the inputs. For example, if you enter side length in centimeters, the area will be in square centimeters (cm²).

Can a regular polygon have any number of sides?

Yes, any integer n ≥ 3 defines a regular n-gon. As n increases, the regular polygon becomes closer in shape to a circle, and the area formula approaches that of a circle.

Regular Polygon Area Calculator – Area, Perimeter & Apothem

How to find the area of a regular polygon

A regular polygon has all sides equal and all interior angles equal. Common examples are the equilateral triangle, square, regular pentagon, and regular hexagon.

There are several equivalent formulas for the area of a regular polygon with n sides:

1. From side length \(s\)

\[ A = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)} \]

2. From perimeter \(P\) and apothem \(a\)

\[ A = \frac{P \cdot a}{2} \]

3. From circumradius \(R\)

\[ A = \frac{n R^2}{2} \sin\left(\frac{2\pi}{n}\right) \]

Key quantities of a regular polygon

n – number of sides (integer, \(n \ge 3\)).
s – side length.
P – perimeter: \(P = n \cdot s\).
a – apothem: distance from the center to the midpoint of a side.
R – circumradius: distance from the center to a vertex.
\(\theta\) – central angle of one sector: \(\theta = \frac{2\pi}{n}\).

Deriving the area formula from triangles

A regular polygon can be split into n congruent isosceles triangles with vertex at the center. Each triangle has:

base \(s\) (a side of the polygon),
height \(a\) (the apothem),
central angle \(\theta = \frac{2\pi}{n}\).

The area of one triangle is \[ A_{\triangle} = \frac{1}{2} \cdot s \cdot a. \] Since there are \(n\) such triangles, \[ A = n \cdot A_{\triangle} = n \cdot \frac{1}{2} s a = \frac{P \cdot a}{2}, \] because \(P = n s\).

Relating side, apothem, and radius

Consider one of the isosceles triangles and drop a perpendicular from the center to the base. This splits it into two right triangles with:

hypotenuse \(R\),
one leg \(a\),
other leg \(\frac{s}{2}\),
angle at the center \(\frac{\theta}{2} = \frac{\pi}{n}\).

From trigonometry:

\[ \tan\left(\frac{\pi}{n}\right) = \frac{\frac{s}{2}}{a} \quad\Rightarrow\quad a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)} \]

\[ \sin\left(\frac{\pi}{n}\right) = \frac{\frac{s}{2}}{R} \quad\Rightarrow\quad R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \]

Substituting \(a\) into \(A = \frac{P a}{2}\) with \(P = n s\) gives \[ A = \frac{n s}{2} \cdot \frac{s}{2 \tan\left(\frac{\pi}{n}\right)} = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)}. \]

Units of area

Whatever unit you use for side length, apothem, or radius (meters, centimeters, inches, feet, etc.), the area is in squared units:

m → m²
cm → cm²
in → in²
ft → ft²

Worked examples

Example 1 – Area of a regular hexagon from side length

Suppose you have a regular hexagon (\(n = 6\)) with side length \(s = 4\ \text{cm}\).

Compute \(\frac{\pi}{n} = \frac{\pi}{6} = 30^\circ\).
\(\tan\left(\frac{\pi}{6}\right) = \tan 30^\circ \approx 0.57735.\)
Use \[ A = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)} = \frac{6 \cdot 4^2}{4 \cdot 0.57735} = \frac{6 \cdot 16}{2.3094} \approx 41.57\ \text{cm}^2. \]

Example 2 – Area of a regular octagon from apothem and perimeter

A regular octagon (\(n = 8\)) has perimeter \(P = 40\ \text{m}\) and apothem \(a = 4.8\ \text{m}\).

Use the formula \(A = \frac{P a}{2}\).
\[ A = \frac{40 \cdot 4.8}{2} = \frac{192}{2} = 96\ \text{m}^2. \]

Example 3 – Area of a regular pentagon from radius

A regular pentagon (\(n = 5\)) has circumradius \(R = 10\ \text{cm}\).

\(\frac{2\pi}{n} = \frac{2\pi}{5} = 72^\circ\).
\(\sin\left(\frac{2\pi}{5}\right) = \sin 72^\circ \approx 0.95106.\)
\[ A = \frac{n R^2}{2} \sin\left(\frac{2\pi}{n}\right) = \frac{5 \cdot 10^2}{2} \cdot 0.95106 = \frac{500}{2} \cdot 0.95106 = 250 \cdot 0.95106 \approx 237.77\ \text{cm}^2. \]

Frequently asked questions

What is the difference between apothem and radius?

The apothem is the distance from the center to the midpoint of a side, while the circumradius is the distance from the center to a vertex. In a regular polygon, both are constant but not equal (except in a square, where \(a = R / \sqrt{2}\)).

Does the formula work for triangles and squares?

Yes. For \(n = 3\) (equilateral triangle), the formula simplifies to \[ A = \frac{\sqrt{3}}{4} s^2. \] For \(n = 4\) (square), it simplifies to \[ A = s^2. \]

What happens as the number of sides goes to infinity?

As \(n \to \infty\), a regular \(n\)-gon with fixed circumradius \(R\) approaches a circle. The area formula \[ A = \frac{n R^2}{2} \sin\left(\frac{2\pi}{n}\right) \] tends to the area of a circle, \(\pi R^2\), because \(\sin x \approx x\) for small \(x\).