What is a regular polygon and how does it differ from an irregular polygon?

A regular polygon has all sides equal in length and all interior angles equal in measure. An irregular polygon lacks at least one of these properties. A square and an equilateral triangle are both regular polygons; a rectangle is irregular because its sides are not all the same length even though its angles are equal. This strict uniformity allows a single formula to compute the area from just the number of sides and one side measurement.

How do you calculate the area of a regular polygon using only its side length?

Apply the formula A = (1/4) x n x s-squared x cot(pi/n), where n is the number of sides and s is the side length. For a regular pentagon (n = 5) with side length 6 cm, the calculation is (1/4) x 5 x 36 x cot(36 degrees) = 45 x 1.3764, giving approximately 61.94 cm squared. No apothem or circumradius measurement is needed; the formula derives all internal geometry from n and s alone.

What is the area of a regular hexagon with a side length of 10 cm?

A regular hexagon with a side length of 10 cm has an area of approximately 259.81 cm squared. The calculation is A = (1/4) x 6 x 100 x cot(30 degrees) = 150 x 1.7321 = 259.81 cm squared. This result also equals (3 times root 3 divided by 2) x 100, confirming that the general regular polygon formula and the hexagon-specific simplified formula produce identical results for all valid inputs.

Why does the regular polygon area formula use the cotangent function?

The cotangent appears because it expresses the apothem, the perpendicular distance from the polygon's center to the midpoint of any side. The apothem equals (s/2) x cot(pi/n). Multiplying the apothem by the full perimeter (n x s) and dividing by 2 gives total enclosed area. Cotangent naturally captures the adjacent-to-opposite ratio in the right triangle formed inside each of the n congruent triangular slices that compose the polygon.

What is the minimum number of sides a regular polygon can have?

The minimum is 3 sides, which forms an equilateral triangle. A two-sided figure cannot enclose any area on a flat plane, making 3 the geometric lower bound for all regular polygons. The formula is fully valid at n = 3: for s = 4 cm, it gives (1/4) x 3 x 16 x cot(60 degrees) = 12 x 0.5774, approximately 6.93 cm squared, which matches the standard equilateral triangle formula (root 3 divided by 4) x s-squared exactly.

How does increasing the number of sides affect a regular polygon's area when side length stays fixed?

Increasing the number of sides increases the total area for a fixed side length, because each additional side extends the perimeter and expands the enclosed region. A regular triangle with s = 5 has an area of about 10.83 square units, a regular hexagon with the same side length reaches about 64.95, and a 12-sided dodecagon reaches approximately 279.90. As n grows toward infinity, the shape converges toward a circle and the area grows without bound.

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Regular Polygon Area Calculator

Compute the area of any regular polygon using A = (n × s²/4) × cot(π/n). Enter number of sides and side length for an instant, accurate result.

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Side Length

Polygon Area

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Polygon Area—square units

The formula

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Regular Polygon Area Formula and Methodology

A regular polygon is a closed, two-dimensional figure with n sides of equal length and n interior angles of equal measure. The regular polygon area calculator computes the enclosed area precisely from just two values: the number of sides and the length of one side. It applies a trigonometric formula trusted by engineers, architects, and mathematicians alike.

The Core Formula

The area A of a regular polygon with n sides, each of length s, is:

A = (n × s²) / 4 × cot(π / n)

Here, cot is the cotangent function — the reciprocal of tangent — and π ≈ 3.14159. This expression is listed as a standard engineering reference in the HP 35s Scientific Calculator User Guide and is rigorously derived in Area and Volume: Where Do the Formulas Come From?, which traces its roots to classical triangle decomposition and the apothem relationship.

Variable Definitions

n — Number of Sides: A positive integer of 3 or greater. Common values include 3 (equilateral triangle), 4 (square), 5 (regular pentagon), 6 (regular hexagon), and 8 (regular octagon). As n increases toward infinity, the polygon approaches a circle.
s — Side Length: The length of any one side, measured in a consistent unit (centimeters, inches, meters, feet). Because all sides of a regular polygon are identical, a single measurement fully defines the figure.

Formula Derivation: From Triangles to the Full Expression

Drawing line segments from the center of the polygon to each vertex divides the figure into n congruent isosceles triangles. Each triangle has a base of length s and a height equal to the apothem — the perpendicular distance from the center to the midpoint of a side. The apothem equals:

a = s / (2 × tan(π / n))

The area of one triangle is (1/2) × s × a = s² / (4 × tan(π / n)). Multiplying by n triangles and substituting cot(θ) = 1 / tan(θ) yields the standard formula. Khan Academy’s lesson on the area of a regular polygon given side length walks through this derivation step by step with interactive diagrams.

Worked Examples

Equilateral Triangle (n = 3, s = 10 cm): A = (3 × 100) / 4 × cot(60°) = 75 × 0.5774 ≈ 43.30 cm²
Square (n = 4, s = 10 cm): A = (4 × 100) / 4 × cot(45°) = 100 × 1 = 100 cm² — confirms the familiar s² rule.
Regular Hexagon (n = 6, s = 10 cm): A = (6 × 100) / 4 × cot(30°) = 150 × 1.7321 ≈ 259.81 cm²
Regular Octagon (n = 8, s = 5 m): A = (8 × 25) / 4 × cot(22.5°) = 50 × 2.4142 ≈ 120.71 m²

Real-World Applications

The regular polygon area formula serves professionals across multiple disciplines:

Architecture and Interior Design: Sizing hexagonal or octagonal floor tiles, windows, and decorative panels to minimize gaps and waste.
Mechanical Engineering: Calculating the cross-sectional area of hex bolts, nuts, and sockets (n = 6) for material and torque specifications.
Urban Planning: Estimating the ground footprint of polygonal plazas, parks, and traffic roundabouts.
Game Development: Computing hex-grid cell areas for terrain scaling in strategy and role-playing games.
Manufacturing: Determining stock material size for polygonal machined components to minimize material waste and cost.

Limitations and Key Notes

The formula applies only to perfect regular polygons where all sides and all angles are equal. For irregular polygons, use the Shoelace Formula (Surveyor’s Formula) instead. As n grows very large, the polygon converges to a circle, and the area approaches πr² where r is the circumradius. Always verify that all side-length values use the same unit; the calculated area will be in the square of that unit.

Reference

Frequently asked questions

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

The area of a regular polygon is A = (n × s²) / 4 × cot(π / n), where n is the number of sides and s is the side length. For example, a regular hexagon with a side length of 6 cm has an area of (6 × 36) / 4 × cot(30°) = 54 × 1.7321 ≈ 93.53 cm². The formula works for any regular polygon with 3 or more sides and reduces to familiar special cases for triangles and squares.

How do you calculate the area of a regular hexagon with a 10 cm side length?

Apply the formula A = (n × s²) / 4 × cot(π / n) with n = 6 and s = 10 cm. This gives A = (6 × 100) / 4 × cot(30°) = 150 × 1.7321 ≈ 259.81 cm². For hexagons, cot(30°) equals √3, so the formula simplifies to A = (3√3 / 2) × s². This identity is widely used in tiling design, honeycomb structure analysis, and hex-bolt engineering calculations.

What is the apothem of a regular polygon and why does it matter for area calculations?

The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side, calculated as s / (2 × tan(π / n)). Area can also be expressed as (1/2) × perimeter × apothem. For a regular pentagon with s = 8 cm, the apothem is 8 / (2 × tan(36°)) ≈ 5.51 cm, giving area = (1/2) × 40 × 5.51 ≈ 110.11 cm². The cotangent formula consolidates the apothem relationship into a single compact expression.

Can the regular polygon area calculator handle equilateral triangles and squares?

Yes. Setting n = 3 computes the area of an equilateral triangle: for s = 8 cm, A = (3 × 64) / 4 × cot(60°) = 48 × 0.5774 ≈ 27.71 cm², which is identical to the result from (√3 / 4) × s². Setting n = 4 reproduces the square area formula: A = (4 × s²) / 4 × cot(45°) = s² × 1 = s². The general formula is fully consistent with all specialized regular polygon cases.

How does the area of a regular polygon change as the number of sides increases?

As n increases, the polygon more closely resembles a circle and its area grows toward the equivalent circular area. With a fixed side length of 1 unit: a triangle (n = 3) gives 0.433 sq units, a hexagon (n = 6) gives 2.598 sq units, and a 100-sided polygon gives approximately 7.954 sq units, approaching the area of the circumscribed circle. This convergence forms the mathematical foundation for classical methods of approximating π using polygons.

What units does the regular polygon area calculator output?

The calculator outputs area in the square of the unit used for side length input. Entering side length in centimeters produces an answer in square centimeters (cm²); entering meters yields square meters (m²); entering inches yields square inches (in²). No automatic unit conversion is applied, so all measurements must use the same unit before calculating. To convert cm² to m², divide by 10,000; to convert in² to ft², divide by 144.