Area of a Regular Polygon Formula
Geometry Formulas: Area, Volume, Perimeter
What is the formula for the area of a regular polygon?
Answer
Area of a regular polygon
A = (n · a²) / (4 · tan(π / n))
Here n is the number of sides, a is the side (or base). The result comes out in square units.
Where the formula comes from
A = (n · a²) / (4 · tan(π / n))
Split the regular n-gon into n identical triangles from the centre. Each has base a and height a / (2·tan(π/n)) — the apothem — so the whole is n·a²/(4·tan(π/n)). For n = 6 this collapses to the hexagon formula, which is the check that it is right.
Worked examples
Let n = 5, a = 6. Then the area of a regular polygon ≈ 61.94 square units.
Let n = 8, a = 3.5. Then the area of a regular polygon ≈ 59.15 square units.
The common mistake
Applying it to an irregular polygon. This formula assumes every side and every angle is equal; an irregular polygon has to be cut into triangles and added up instead.
Other quantities: Regular polygon
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See Also
- Square Footage Calculator - Width × length in feet to square feet
- Square Feet to Square Meters (Sq Ft to Sq M) - Convert Square Feet to Square Meters
- Square Meters to Square Feet (Sq M to Sq Ft) - Convert Square Meters to Square Feet
- Gravel Calculator - Area to cubic yards, tons and coverage
- Topsoil Calculator - Cubic yards, weight and coverage for any area
- Multiplication Facts - Any two numbers from 1 to 9, with the working shown
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Values from the Formula (a = 6)
| If n = | Area |
|---|---|
| 3 | ≈ 15.59 square units |
| 4 | 36 square units |
| 5 | ≈ 61.94 square units |
| 6 | ≈ 93.53 square units |
| 8 | ≈ 173.82 square units |
| 10 | ≈ 276.99 square units |
| 12 | ≈ 403.06 square units |
Geometry Formulas: Area, Volume, Perimeter
These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the area of a regular polygon? (The answer is: A = (n · a²) / (4 · tan(π / n))). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter