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Regular Polygon Area Calculator

Calculate the area of any regular polygon given the number of sides and side length using A = n*s^2/(4*tan(PI/n)). Works for equilateral triangles, squares, pentagons, hexagons, and any regular n-gon used in architecture, tiling, and engineering.

Reviewed by Christopher FloiedUpdated

This free online regular polygon area calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Number of equal sides (minimum 3)

Length of each side

Results

Area

64.9519 sq units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Regular Polygon Area Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Regular Polygon Area Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Regular Polygon Area Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Regular Polygon Area Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Regular Polygon Area Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the area of any regular polygon given the number of sides and side length using A = n*s^2/(4*tan(PI/n)). Works for equilateral triangles, squares, pentagons, hexagons, and any regular n-gon used in architecture, tiling, and engineering. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Regular Polygon Area Calculator

The Regular Polygon Area Calculator computes the area of any regular polygon (equilateral triangle, square, pentagon, hexagon, etc.) given the number of sides and the side length. The formula A = n*s^2/(4*tan(pi/n)) unifies all regular polygon areas into a single expression. As the number of sides increases, a regular polygon approaches a circle, and this formula smoothly transitions to pi*r^2 in the limit. Regular polygons are ubiquitous in nature (honeycomb hexagons), architecture (octagonal towers), engineering (hex bolt heads), and design (pentagonal stars). This calculator handles any regular polygon from triangles to 100-gons.

The Math Behind It

A regular polygon has n sides of equal length s and n equal interior angles. It can be decomposed into n isosceles triangles by drawing lines from the center to each vertex. Each triangle has base s and height a (the apothem, the perpendicular distance from the center to a side). The area of each triangle is (1/2)*s*a, so the total area is (n/2)*s*a. The apothem a relates to the side length by a = s/(2*tan(pi/n)). Substituting: A = (n/2)*s*s/(2*tan(pi/n)) = n*s^2/(4*tan(pi/n)). This formula works for all n >= 3. For specific cases: n=3 (equilateral triangle) gives A = (sqrt(3)/4)*s^2; n=4 (square) gives A = s^2; n=6 (regular hexagon) gives A = (3*sqrt(3)/2)*s^2. The circumradius R = s/(2*sin(pi/n)) and the inradius (apothem) a = s/(2*tan(pi/n)) are also useful. As n approaches infinity, the regular polygon approaches a circle. Using the limit of n*s^2/(4*tan(pi/n)) as n->inf with perimeter P = n*s held constant, you can show the area approaches P^2/(4*pi) = pi*r^2, confirming the circle area formula. This limit argument was used by Archimedes to approximate pi by inscribing and circumscribing regular 96-gons around a circle. Regular polygons that tile the plane are limited to triangles (n=3), squares (n=4), and hexagons (n=6), a result proven by examining the interior angles. Semi-regular tilings combine different regular polygons. In nature, hexagonal tiling appears in beehives because hexagons maximize area for a given perimeter in a tiling arrangement.

Formula Reference

Regular Polygon Area

A = n * s^2 / (4 * tan(pi/n))

Variables: n = number of sides, s = side length

Alternative (using apothem)

A = (1/2) * n * s * a

Variables: a = apothem (distance from center to midpoint of a side)

Worked Examples

Example 1: Regular Hexagon with Side 5

Find the area of a regular hexagon with side length 5 cm.

Step 1:n = 6, s = 5 cm
Step 2:A = 6 * 25 / (4 * tan(pi/6)) = 150 / (4 * 0.5774) = 150 / 2.3094 = 64.95

The area of the regular hexagon is approximately 64.95 square cm.

Example 2: Regular Pentagon with Side 8

Find the area of a regular pentagon with side length 8 m.

Step 1:n = 5, s = 8 m
Step 2:A = 5 * 64 / (4 * tan(pi/5)) = 320 / (4 * 0.7265) = 320 / 2.9062 = 110.11

The area of the regular pentagon is approximately 110.11 square meters.

Common Mistakes & Tips

  • !Using degrees instead of radians in the tangent function. The formula requires tan(pi/n) in radians, not tan(180/n) in degrees.
  • !Confusing the apothem with the circumradius. The apothem goes from the center to the midpoint of a side; the circumradius goes from the center to a vertex.
  • !Applying the formula to irregular polygons. This formula only works when all sides and all angles are equal.
  • !Forgetting that the formula gives area in square units of whatever unit the side length is measured in.

Related Concepts

Parallelogram Area

Regular polygons with n=4 are squares, a special case of parallelograms where A = s^2 = b*h.

Semicircle Area

As n increases, regular polygons approach circles. The semicircle area formula represents half of this limiting case.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

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Frequently Asked Questions

What happens as the number of sides approaches infinity?

The regular polygon approaches a circle. The area formula n*s^2/(4*tan(pi/n)) approaches pi*r^2 in the limit, where r is the circumradius. This is how Archimedes approximated pi over 2000 years ago.

Which regular polygons can tile the plane?

Only three: equilateral triangles (n=3), squares (n=4), and regular hexagons (n=6). These are the only regular polygons whose interior angles evenly divide 360 degrees, allowing them to fit together without gaps.

How do I find the area if I know the apothem instead of the side length?

Use A = (1/2)*perimeter*apothem = (1/2)*n*s*a. If you only know the apothem a and n, first find s = 2*a*tan(pi/n), then compute the area.