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

For a regular polygon with n sides each of length s, the area is A = (n x s^2) / (4 x tan(pi/n)). Equivalently, if you know the apothem a (the perpendicular distance from the centre to a side), the area is A = (1/2) x perimeter x a = (1/2) x (n x s) x a. If you know the circumradius R (distance from centre to a vertex), the area is A = (1/2) x n x R^2 x sin(2pi/n). All three formulas give the same result when the same polygon is described.

What is the difference between the apothem and the circumradius of a polygon?

The apothem (also called the inradius) is the perpendicular distance from the centre of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle. The circumradius is the distance from the centre to any vertex (corner), and is the radius of the circumscribed circle. For a given polygon, the circumradius is always larger than the apothem. For a regular hexagon with side length s, the apothem is s x sqrt(3)/2 and the circumradius equals s.

How do I find the area of a regular hexagon?

For a regular hexagon (6 sides) with side length s, the area formula simplifies to A = (3 x sqrt(3) / 2) x s^2. For example, a hexagon with side length 10 cm has area = (3 x 1.73205 / 2) x 100 = 2.59808 x 100 = 259.81 cm^2 (rounded to 2 decimal places). This matches the general formula A = (n x s^2) / (4 x tan(pi/n)) with n = 6.

Area of a Regular Polygon Calculator

This calculator finds the area of any regular polygon, a shape with equal sides and equal angles, from a single known measurement. Choose the number of sides from the dropdown, ranging from an equilateral triangle (3 sides) up to a 20-sided icosagon, or enter a custom number of sides above 3. Pick your unit of measurement (millimetres, centimetres, metres, inches or feet), then choose which measurement you know: the side length, the apothem (the distance from the centre to the midpoint of a side, also called the inradius), or the circumradius (the distance from the centre to a corner). Enter that one figure and the calculator instantly returns the area, the perimeter, the apothem and the circumradius, along with the polygon's name and interior angle. It also shows the exact formula being used and a full worked example, stepping through tan(pi/n), the numerator and denominator, so you can see exactly how the answer was reached rather than just reading off a number. This is useful for maths homework, trade and construction jobs like laying pavers or fencing a hexagonal deck, craft and design projects, or checking a textbook answer. Results are calculated to full floating-point precision and displayed rounded to four decimal places, so round further to suit the accuracy of your own measurements.

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Standard formula Euclidean geometry. Valid for all convex regular polygons with 3 or more sides.

1. Polygon Shape

Number of sides

Custom number of sides (3 or more)

Unit of measurement

2. Known Measurement

Input method

Side length (s)

Please enter a positive number.

Formula in use A = (n x s²) / (4 x tan(π/n))

Results

Area
-
-

Perimeter

Apothem (a)

Circumradius (R)

Polygon Dimensions

Number of sides (n)-

Polygon name-

Side length (s)-

Apothem (a)-

Circumradius (R)-

Interior angle-

Area-

Worked Example

n (sides)-

Input-

Side length s-

tan(π/n)-

n x s²-

4 x tan(π/n)-

Area = n x s² / (4 x tan(π/n))-

How to Calculate the Area of a Regular Polygon

A regular polygon has all sides the same length and all interior angles equal. Common examples include the equilateral triangle (3 sides), square (4 sides), regular pentagon (5 sides), regular hexagon (6 sides), and regular octagon (8 sides). As the number of sides increases, a regular polygon approaches the shape of a circle.

The Three Main Formulas

There are three equivalent formulas depending on which measurement you know:

Known measurement	Formula	Notes
Side length (s)	A = (n x s²) / (4 x tan(π/n))	Most common starting point
Apothem (a)	A = n x a² x tan(π/n)	Apothem is also called the inradius
Circumradius (R)	A = (1/2) x n x R² x sin(2π/n)	Circumradius is the distance to a vertex

In all formulas, n is the number of sides. The apothem a is the perpendicular distance from the centre to the midpoint of a side. The circumradius R is the distance from the centre to any vertex (corner).

Relationship Between s, a and R

Given side length s, you can find the apothem and circumradius:

Apothem: a = s / (2 x tan(π/n))
Circumradius: R = s / (2 x sin(π/n))

Conversely, given the apothem a, the side length is s = 2 x a x tan(π/n), and from circumradius R the side length is s = 2 x R x sin(π/n).

Interior Angles

Each interior angle of a regular polygon with n sides is ((n - 2) x 180) / n degrees. For example, a regular hexagon has interior angles of (4 x 180) / 6 = 120 degrees each. The sum of all interior angles is (n - 2) x 180 degrees.

Common Regular Polygons

Polygon	Sides (n)	Interior angle	Area formula (side s)
Equilateral triangle	3	60°	A = (s² x √3) / 4
Square	4	90°	A = s²
Pentagon	5	108°	A = (s² / 4) x √(25 + 10√5)
Hexagon	6	120°	A = (3√3 / 2) x s²
Octagon	8	135°	A = 2(1 + √2) x s²

Worked Example: Regular Hexagon, Side = 10 cm

Using the side length formula with n = 6, s = 10 cm:

Calculate π/n = π/6 = 0.52360 radians
Calculate tan(π/6) = tan(30°) = 0.57735
Numerator: n x s² = 6 x 100 = 600
Denominator: 4 x tan(π/6) = 4 x 0.57735 = 2.30940
Area = 600 / 2.30940 = 259.81 cm²

The apothem of this hexagon is a = 10 / (2 x tan(π/6)) = 10 / 1.15470 = 8.66 cm. The circumradius equals the side length for a regular hexagon: R = 10 cm.

Related Calculators

Sources and method: Standard Euclidean geometry. Formulas consistent with NZQA Level 2 and Level 3 Mathematics (AS91261, AS91587). See also: Weisstein, E.W. "Regular Polygon" from MathWorld (mathworld.wolfram.com/RegularPolygon.html).

Results are computed to full floating-point precision and displayed rounded to 4 decimal places. For practical applications, round to the precision appropriate for your measurement accuracy.