Polygon area formulas (regular polygons)

A regular polygon is a polygon where every side length is the same and every interior angle is the same. Because the shape is perfectly “even,” we can break it into n identical triangles that all meet at the center. That triangle idea is the secret behind every polygon area formula.

If you know the number of sides n and the side length s, a widely used formula is:

• Area = (n · s²) / (4 · tan(π/n))

Why does tan(π/n) show up? Because when you connect the center to two neighboring vertices, you create an isosceles triangle. Half of that triangle is a right triangle where tan links the apothem (adjacent) and half-side (opposite).

This is the “cleanest” formula and also one of the most practical:

• Area = (1/2) · P · a

Think of it like this: the polygon is n triangles. Each triangle has base s and height a, so its area is (1/2)·s·a. Multiply by n and you get (1/2)·(n·s)·a. But n·s is the perimeter P.

If you know the distance from the center to any vertex (the circumradius), you can use:

• Area = (1/2) · n · R² · sin(2π/n)

Here, 2π/n is the central angle (in radians) of each triangle at the center. The triangle area formula (1/2)ab sin(C) becomes (1/2)·R·R·sin(2π/n), repeated n times.

Step-by-step examples

Examples are where polygon area finally clicks. Below are real, “calculator-style” walkthroughs using each method. You can plug the same numbers into the calculator above to confirm.

• Formula: Area = (n·s²) / (4·tan(π/n))
• Compute: π/n = π/6 = 30° (≈ 0.5236 rad)
• tan(π/6) ≈ 0.57735
• Area ≈ (6·64) / (4·0.57735) = 384 / 2.3094 ≈ 166.28

• Formula: Area = (1/2)·P·a
• Area = 0.5 · 80 · 12 = 480

• Formula: Area = (1/2)·n·R²·sin(2π/n)
• 2π/n = 2π/5 = 72° (≈ 1.2566 rad)
• sin(72°) ≈ 0.95106
• Area ≈ 0.5 · 5 · 100 · 0.95106 = 250 · 0.95106 ≈ 237.76

If your answers feel “too big” or “too small,” double-check your units and whether you entered apothem vs radius. Apothem goes to a side (center-to-edge). Radius goes to a vertex (center-to-corner).

What this calculator is doing behind the scenes

When you tap Calculate Area, the calculator follows a simple pattern: it validates inputs, chooses the correct formula based on your selected method, and then computes area plus a few “nice-to-have” outputs like perimeter, apothem, and side length (when they can be derived).

The math is based on splitting the polygon into n congruent triangles. Each triangle has: (1) a central angle of 2π/n, (2) two equal sides (radius to vertices), and (3) a base equal to the side length.

• Area: main result (square units).
• Perimeter: either computed as n·s or echoed from your input.
• Apothem: computed from side length when possible: a = s / (2·tan(π/n)).
• Side length: computed from perimeter: s = P/n.

The preview uses your n value and draws a regular polygon on a canvas by placing points equally around a circle. This makes it easier to catch “oops” moments — like accidentally typing 12 sides when you meant 6.

Polygon area questions people ask all the time

• Does this work for irregular polygons?

  Not directly. Irregular polygons don’t have a single side length or angle that repeats. For irregular shapes, you typically need coordinates (shoelace formula), triangulation, or a CAD measurement. This calculator is specifically for regular polygons.

• What’s the difference between apothem and radius?

  The apothem goes from the center to the middle of a side (center-to-edge). The circumradius goes from the center to a vertex (center-to-corner). They’re related, but they’re not the same measurement.

• Why is the answer in “square units”?

  Area measures 2D space. If your side length is in meters, the area is in square meters (m²). If your side is inches, the area is square inches (in²). That’s why unit choice matters.

• What is n allowed to be?

  Any whole number n ≥ 3. A polygon needs at least 3 sides. As n gets very large, a regular polygon starts to look like a circle (and the area approaches the circle’s area).

• How do I find the area of a hexagon?

  If it’s a regular hexagon, you can use this calculator with n = 6 and either side length, perimeter + apothem, or circumradius. A fun shortcut: a regular hexagon can be split into 6 equilateral triangles.

• Can I use perimeter only?

  Not for area. Perimeter alone doesn’t “lock in” the shape’s size. You need another piece of information, like apothem, side length, or radius.

• What’s the easiest formula to remember?

  Area = (1/2) · P · a. If you can measure or compute apothem, it’s clean, fast, and widely used.

Where polygon area shows up in real life

Polygon area isn’t just a school topic — it pops up in surprising places. The “regular polygon” case is especially common because designers like symmetry and repeating patterns.

• Architecture: tiled floors, decorative patterns, octagonal features, pavilion designs.
• DIY / construction: estimating material for a symmetric patio, gazebo base, or tabletop.
• Graphic design: badges, icons, game assets, and UI components that use regular polygons.
• 3D printing / CNC: polygon outlines for parts, mounts, plates, and custom shapes.
• Board games: hex grids and regular polygons for map tiles.

Want “viral” practice? Challenge friends: who can guess the area of a hexagon or octagon fastest — then verify here.

How to avoid wrong answers

• Using degrees inside trig: This calculator uses radians internally (π/n). You don’t have to convert.
• Mixing up apothem and radius: Apothem hits the side; radius hits the vertex.
• Non-regular shapes: If sides aren’t equal, regular formulas won’t match reality.
• Forgetting units: Area is squared. If you choose “ft” you’ll get ft².
• n must be an integer: “6.5 sides” can’t exist — the calculator enforces whole numbers.