Area Of Regular Polygon Calculator

A regular polygon is a polygon with all sides equal and all interior angles equal — think equilateral triangle, square, regular pentagon, hexagon, and so on. If you need the area of a regular polygon for homework, CAD work, architecture, craft patterns, or geometry checks, the Area of Regular Polygon Calculator gives you a quick, reliable result based on either the side length or the apothem.

Area Of Regular Polygon Calculator

What the calculator does

The calculator accepts either:

  • Number of sides n and side length s, or
  • Number of sides n and apothem a (the distance from center to midpoint of a side).

It returns:

  • Area of the polygon (in the same square units as your inputs),
  • Optional perimeter and apothem (if computed from side length),
  • A step-by-step breakdown so you can verify the math.

Key formulas (plain text)

  1. Using side length s:
Area = (1 / 4) × n × s^2 × cot(π / n)

cot(π / n) is the cotangent of π divided by n (cot = 1 / tan).

  1. Using apothem a and perimeter P:
Area = (1 / 2) × a × P where P = n × s
  1. Apothem from side length:
a = s / (2 × tan(π / n))

All trigonometric functions use radians (π = 3.141592653589793…).

Why these formulas work (brief intuition)

A regular n-gon can be split into n identical isosceles triangles, each with base s and height equal to the apothem a. Area of one triangle is 0.5 × base × height = 0.5 × s × a. Multiply by n gives Area = 0.5 × a × n × s = 0.5 × a × P. Using trigonometry, a can be written in terms of s and n, which yields the cot(π/n) formula.

How to use the calculator (simple steps)

  1. Enter the number of sides n (integer ≥ 3).
  2. Provide either the side length s or the apothem a.
  3. Select the units (meters, inches, cm, etc.).
  4. Click Calculate. The tool outputs the area, and — if you gave s — it also shows the apothem and perimeter.

Worked examples (digit-by-digit accuracy)

Example A — Square (sanity check)

  • n = 4, s = 5 (units: meters)

Square area by familiar formula is s² = 5² = 25 m². Using the general formula:

  1. Compute cot(π/n) = cot(π/4). π/4 = 45°, cot(45°) = 1.
  2. Area = (1/4) × 4 × 5² × 1 = 1 × 25 = 25 m².
    Matches the expected result.

Example B — Regular hexagon (n = 6) with side s = 10

We use the side-based formula:

Area = (1/4) × n × s^2 × cot(π / n)

Step-by-step:

  1. n = 6, s = 10 → s² = 100.
  2. π / n = π / 6 = 30 degrees.
  3. cot(π/6) = cot(30°) = √3 ≈ 1.7320508075688772.
  4. Compute (1/4) × n × s² = 0.25 × 6 × 100 = 150.
  5. Area = 150 × 1.7320508075688772 = 259.8076211353316.

Rounded sensibly: 259.8076 units².

(You can use the calculator to display whatever number of decimal places you prefer.)

Example C — Regular octagon (n = 8) with side s = 6 (using apothem method)

First compute apothem a = s / (2 × tan(π / n)):

  1. n = 8, s = 6.
  2. π / 8 = 22.5°. tan(π/8) = tan(22.5°) ≈ 0.41421356237309503.
  3. 2 × tan(π/8) = 0.82842712474619006.
  4. a = 6 ÷ 0.82842712474619006 = 7.242640687119286.

Perimeter P = n × s = 8 × 6 = 48.

Area = 0.5 × a × P = 0.5 × 7.242640687119286 × 48
First compute a × P = 7.242640687119286 × 48 = 347.647 (exact: 347.647, but more precisely 347.647376...); half of that is 173.82337649086287.

Rounded: 173.8234 units².

Practical tips & common pitfalls

  • Units matter. If s is in cm, the area is cm². Convert units before or after calculation depending on your needs.
  • Use radians for trig functions if doing hand-calc or coding. Most calculators and programming languages accept radians; convert degrees to radians by multiplying degrees × π/180.
  • n must be ≥ 3. n = 3 is a triangle (works fine). n = 4 gives the square special case.
  • Round only at the end. Keep intermediate precision to avoid rounding error, especially for polygons with many sides.
  • Large n approximates a circle. As n increases, the regular polygon approaches the area of a circle with radius equal to its circumradius.
  • If you know apothem a, the (1/2)aP formula is simplest and often most accurate.

Use cases (who benefits)

  • Students solving geometry problems and homework.
  • Architects and designers modeling floor tiles or decorative patterns.
  • CNC and laser-cut hobbyists calculating material area.
  • Game designers or 3D modelers building regular polygons.
  • Civil and structural engineers when quick area checks are needed.

Accuracy & numerical stability

For polygons with a high number of sides (large n), cot(π/n) can be computed using high-precision trig functions or by using the apothem formula (a = s / (2 tan(π/n))) which avoids loss of precision. The calculator uses stable math functions to keep results accurate.

20 Frequently Asked Questions (FAQs)

1. What is a regular polygon?
A polygon with all sides equal and all interior angles equal.

2. Which inputs does the calculator need?
Number of sides n (≥3) and either side length s or apothem a.

3. What units does the area use?
Square units of whatever length units you enter (e.g., meters → m²).

4. Why use cot(π/n) instead of apothem?
Both are equivalent; cot form is compact when you only know s. Apothem form is simpler when a is known.

5. How to compute apothem from side?
a = s / (2 × tan(π / n)).

6. Is there a formula using circumradius R?
Yes: Area = 0.5 × n × R² × sin(2π / n) where R is circumradius.

7. What if I only know perimeter P?
If you know P and n, then s = P / n; then use the usual formulas.

8. Can the calculator handle fractional sides?
No — n must be an integer ≥ 3. Side lengths and apothems can be fractional.

9. What about very large n?
As n → ∞ with fixed circumradius, the polygon’s area approaches the circle’s area.

10. Is the formula valid for concave polygons?
No — only for regular convex polygons.

11. Can I compute area for an equilateral triangle?
Yes — with n = 3. The formula gives Area = (sqrt(3)/4) × s².

12. How accurate is the result?
Very accurate; final precision depends on the trig function precision and rounding preference.

13. Do I need a scientific calculator?
For hand work yes — to evaluate tan or cot. The online calculator does it for you.

14. What if interior angles are not equal?
Then the polygon is not regular and these formulas do not apply.

15. Can I get an exact symbolic answer?
For some n (like 3, 4, 6) exact radicals exist; the calculator can show symbolic forms where available.

16. Does the calculator show perimeter and apothem?
Yes — if you input s, it can compute P = n × s and a = s / (2 tan(π / n)).

17. How do I convert area units?
Multiply by conversion factor squared (e.g., 1 m² = 10000 cm²).

18. Is the formula derived from triangles?
Yes — by dividing the polygon into n congruent isosceles triangles.

19. Can I use the tool for tiling patterns?
Absolutely — it’s ideal for estimating material area in tiling.

20. Is this calculator free?
Yes — the Area of Regular Polygon Calculator is free and instant to use.

Final notes

The Area of Regular Polygon Calculator removes the tedium and potential errors of trig-heavy manual calculation. Whether you’re a student needing homework help, a maker cutting panels, or a designer checking areas, this tool gives fast, reliable areas with transparent formulas and step-by-step outputs.