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Geometry

Regular Polygon Area

A = ¼ n s² cot(π/n). Free online Regular Polygon Area. Calculate regular polygon area online — fast, accurate, mobile-friendly, no signup needed.

Area = ½ · perimeter · apothem.
Area
64.951905

Derivation

  1. ├── 01Givenn = 6, s = 5
  2. ├── 02Formulat × a² / (4 × tan(π / t))
  3. └── 03Compute Area64.951905
Did you know?

Honeybees build hexagonal cells because hexagons tile the plane with minimum perimeter for a given area — a result proved only in 1999 by Thomas Hales.

§01What is

Understanding the Regular Polygon Area

The Regular Polygon Area computes Area from 2 inputs: sides, side length. A = ¼ n s² cot(π/n).

Geometry is what turns raw measurements into useful answers about space — how much paint, how big a yard, how much material a project will need. Every craftsperson, architect, and DIYer reaches for these formulas regularly. The Regular Polygon Area sits in that toolkit — it A = ¼ n s² cot(π/n). Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t × a² / (4 × tan(π / t))

Where

n
Sides
s
Side length
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Sides = 6, Side length = 5.

  1. 01Start by noting the input — Sides: 6.
  2. 02Start by noting the input — Side length: 5.
  3. 03Substitute these values into the formula: t × a² / (4 × tan(π / t))
  4. 04Compute Area: the calculator returns 64.9519.
  5. 05Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Regular Polygon Area Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Sides halved

n = 3 (from 6)

Keep every other input at its default and halve the sides. See how area responds.

  1. 01New Sides: 3
  2. 02Baseline Area: 64.9519
  3. 03New Area: 10.8253
  4. 04Area decreases by 83.3% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Sides doubled

n = 12 (from 6)

Keep every other input at its default and double the sides. See how area responds.

  1. 01New Sides: 12
  2. 02Baseline Area: 64.9519
  3. 03New Area: 279.904
  4. 04Area increases by 330.9% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Side length halved

s = 2.5 (from 5)

Keep every other input at its default and halve the side length. See how area responds.

  1. 01New Side length: 2.5
  2. 02Baseline Area: 64.9519
  3. 03New Area: 16.238
  4. 04Area decreases by 75% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Side length doubled

s = 10 (from 5)

Keep every other input at its default and double the side length. See how area responds.

  1. 01New Side length: 10
  2. 02Baseline Area: 64.9519
  3. 03New Area: 259.808
  4. 04Area increases by 300% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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