Area of N Gons Calculator

Calculate the area of any n-sided polygon (n-gon) with our precise online calculator. Whether you're working with regular polygons like pentagons or hexagons, or irregular polygons with known side lengths and angles, this tool provides accurate results with clear formulas and examples.

How to Use This Calculator

Using our area of n-gons calculator is simple:

Enter the number of sides (n) for your polygon.
For regular polygons, enter the side length. For irregular polygons, enter the side lengths separated by commas.
Select the appropriate unit of measurement (square meters, square feet, etc.).
Click "Calculate" to get the area.
Review the result and formula used.

The calculator handles both regular and irregular polygons. For regular polygons, it uses the standard formula. For irregular polygons, it uses the shoelace formula when coordinates are provided, or calculates based on side lengths and angles when available.

Formula Explained

The area of a regular n-gon with side length s is calculated using:

Area = (n × s²) / (4 × tan(π/n))

For irregular polygons, the shoelace formula is used when coordinates are provided:

Area = 1/2 |Σ(xi × yi+1) - Σ(yi × xi+1)| where (xi,yi) are vertices

When only side lengths and angles are known, the area can be calculated by dividing the polygon into triangles and summing their areas.

Note: For irregular polygons, ensure you provide either all coordinates or all side lengths and angles for accurate results.

Worked Examples

Example 1: Regular Pentagon

Calculate the area of a regular pentagon with side length 5 cm.

Using the formula:

Area = (5 × 5²) / (4 × tan(π/5)) ≈ 55.258 cm²

Example 2: Irregular Quadrilateral

Calculate the area of an irregular quadrilateral with vertices at (0,0), (4,0), (4,3), and (0,3).

Using the shoelace formula:

Area = 1/2 |(0×0 + 4×3 + 4×3 + 0×0) - (0×4 + 0×4 + 3×0 + 3×0)| = 12 cm²

Frequently Asked Questions

What is the difference between regular and irregular polygons?

Regular polygons have all sides and angles equal, while irregular polygons have sides and angles of different measures.

Can I calculate the area of a polygon with missing information?

Yes, but you'll need to provide either all coordinates, all side lengths and angles, or enough information to derive the missing values.

What units should I use for the side lengths?

Use consistent units for all side lengths. The calculator will display the area in the corresponding square units.

Is this calculator accurate for very large polygons?

Yes, the calculator uses precise mathematical formulas and handles large numbers accurately.