N-Gon Calculator
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Reviewed by Calculator Editorial Team
An n-gon calculator helps you determine the properties of regular polygons with any number of sides. Whether you're studying geometry, designing patterns, or analyzing shapes, this tool provides quick calculations for perimeter, area, side length, and more.
What is an N-gon?
An n-gon is a polygon with n sides, where n is an integer greater than or equal to 3. Regular n-gons have all sides and angles equal. Common examples include triangles (3-gons), squares (4-gons), pentagons (5-gons), and hexagons (6-gons).
Regular polygons are symmetric and have equal side lengths and angles. Irregular polygons have sides and angles of different measures.
Properties of Regular N-gons
- Perimeter: The sum of all side lengths.
- Area: The space enclosed by the polygon.
- Side Length: The length of each side.
- Central Angle: The angle subtended by one side at the center.
- Interior Angle: The angle between two adjacent sides.
How to Use the Calculator
To calculate properties of an n-gon:
- Enter the number of sides (n) of the polygon.
- Select whether you're providing the perimeter or side length.
- Enter the corresponding value.
- Click "Calculate" to see the results.
If you know the perimeter (P) and number of sides (n), the side length (s) is calculated as:
s = P / n
The calculator will display the perimeter, area, side length, central angle, and interior angle of the regular n-gon.
Formulas
The following formulas are used in the n-gon calculator:
Perimeter
P = n × s
Where:
- P = Perimeter
- n = Number of sides
- s = Side length
Area
A = (n × s²) / (4 × tan(π/n))
Where:
- A = Area
- n = Number of sides
- s = Side length
- π = Pi (approximately 3.14159)
Central Angle
θ = 360° / n
Where:
- θ = Central angle
- n = Number of sides
Interior Angle
α = (n - 2) × 180° / n
Where:
- α = Interior angle
- n = Number of sides
Examples
Let's calculate properties for a regular hexagon (6-gon) with a perimeter of 36 units.
Example Calculation
Given:
- n = 6
- P = 36 units
Side length (s):
s = P / n = 36 / 6 = 6 units
Area (A):
A = (6 × 6²) / (4 × tan(π/6)) ≈ (6 × 36) / (4 × 0.577) ≈ 216 / 2.309 ≈ 93.53 square units
Central angle (θ):
θ = 360° / 6 = 60°
Interior angle (α):
α = (6 - 2) × 180° / 6 = 4 × 30° = 120°
For a regular pentagon (5-gon) with a side length of 5 units:
Example Calculation
Given:
- n = 5
- s = 5 units
Perimeter (P):
P = 5 × 5 = 25 units
Area (A):
A = (5 × 5²) / (4 × tan(π/5)) ≈ (5 × 25) / (4 × 0.7265) ≈ 125 / 2.906 ≈ 43.01 square units
Central angle (θ):
θ = 360° / 5 = 72°
Interior angle (α):
α = (5 - 2) × 180° / 5 = 3 × 36° = 108°
FAQ
- What is the difference between a regular and irregular n-gon?
- A regular n-gon has all sides and angles equal, while an irregular n-gon has sides and angles of different measures.
- Can I calculate properties for non-integer values of n?
- No, n must be an integer greater than or equal to 3. Polygons with fractional sides are not standard geometric shapes.
- What is the maximum number of sides an n-gon can have?
- There is no theoretical maximum, but practical applications typically use n ≤ 100 for regular polygons.
- How accurate are the calculations in this calculator?
- The calculator uses standard geometric formulas and JavaScript's built-in trigonometric functions, providing accurate results for regular polygons.
- Can I use this calculator for irregular polygons?
- No, this calculator is designed for regular polygons only. For irregular polygons, you would need to know all side lengths and angles to calculate properties.