Regular Polygon Area Calculator

Regular Polygon Area Formulas

Three practical formulas let you calculate area depending on which measurement you know. All apply to regular polygons only—those with equal sides and equal interior angles.

• n — Number of sides
• a — Length of one side
• r — Circumradius (distance from center to any vertex)
• ap — Apothem (perpendicular distance from center to the midpoint of any side)
• π — Pi, approximately 3.14159

Understanding Regular Polygons

A regular polygon has all sides of equal length and all interior angles equal. Common examples include equilateral triangles (3 sides), squares (4), regular pentagons (5), and hexagons (6). Irregular polygons—those with unequal sides or angles—require different calculation methods.

The three formulas above correspond to the three most useful measurements you might have:

• Side length (a): The most straightforward input. Measure or know the distance between adjacent vertices.
• Circumradius (r): The radius of a circle that passes through all vertices. Used when the polygon is inscribed in a circle.
• Apothem (ap): The inradius—how far the center sits from any side's midpoint. Essential for polygons where you can measure perpendicular distance to the perimeter.

Each formula yields identical results; choose based on what measurement is available to you.

Practical Calculation Example

Suppose you're designing a decorative tile pattern and need the area of a regular dodecagon (12-sided polygon) with 5-inch sides.

Using the side-length formula:

Area = 12 × 5² ÷ (4 × tan(π/12))
Area ≈ 12 × 25 ÷ (4 × 0.2679)
Area ≈ 279.9 square inches

This dodecagon occupies roughly 1.94 square feet—useful for determining material needed or fitting multiple tiles in a space.

Common Pitfalls and Considerations

Avoid these mistakes when calculating polygon area:

1. Confusing apothem with side length — The apothem is the perpendicular distance from the center to a side's midpoint, not the distance between two vertices. They differ significantly. For a square with 4-inch sides, the apothem is only 2 inches. Using the wrong value produces incorrect results by a factor of 2 or more.
2. Forgetting angle units in trigonometry — The formulas use π/n and 2π/n in radians, not degrees. Most calculators default to degrees. If your tool requires degree input, convert: degrees = (π/n) × (180/π). Many errors stem from this unit mismatch.
3. Assuming the polygon is regular when it isn't — These formulas only work for regular polygons where all sides and angles are equal. If sides vary or angles differ, use the Shoelace formula (splitting into triangles) instead. Visual inspection may deceive—measure carefully.
4. Rounding intermediate steps too early — Store full precision through all calculations, especially when working with trigonometric functions. Rounding tan(π/n) or sin(2π/n) before multiplication introduces creeping error, particularly noticeable in polygons with many sides.

When to Use Each Formula

Side length formula (most common): You've measured the polygon's perimeter or know the edge length directly. This is the default choice for manufactured regular polygons—tiles, bolts, architectural elements.

Circumradius formula (when inscribed in circles): The polygon fits inside a circle, and you know or can measure the circle's radius. Useful in astronomy (regular star arrangements) or when designing objects that must fit within a circular boundary.

Apothem formula (construction and manufacturing): You've measured the perpendicular distance from the center to any flat edge. Common in engineering where this distance is easier to verify than side length, especially for large or irregular assemblies.

In practice, the side-length method dominates because edges are the easiest to measure or specify in plans and blueprints.