Polygon Area Calculator
Calculate polygon area using multiple mathematical methods
Input Values
Result
Regular polygon formula: A = (n × s²) / (4 × tan(π/n))
Polygon Area Methods
Method 1: Regular Polygon Formula
For regular polygons (equal sides and angles), use: A = (n × s²) / (4 × tan(π/n))
This formula works by dividing the polygon into n congruent isosceles triangles.
Method 2: Apothem Method
If you know the apothem: A = (1/2) × perimeter × apothem
The apothem is the perpendicular distance from center to any side.
Method 3: Shoelace Formula
For any polygon with known coordinates: A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
Works for irregular polygons. Also called the surveyor's formula.
Frequently Asked Questions
Which method should I use?
Use the regular polygon formula for equal-sided polygons. Use the shoelace formula for irregular polygons with known coordinates. Use the apothem method when you know the apothem value.
What is the shoelace formula?
The shoelace formula calculates polygon area from vertex coordinates by summing cross-products. It's called "shoelace" because the calculation pattern resembles lacing a shoe.
Can I use this for irregular polygons?
Yes, use the shoelace (coordinates) method for irregular polygons. The regular polygon and apothem methods only work for regular polygons.
Why divide by 2 in area formulas?
Many area formulas involve triangulation. Since triangle area = (1/2) × base × height, the factor of 1/2 carries through to polygon formulas.
What if my polygon has a hole?
Calculate the area of the outer polygon and subtract the area of the inner polygon (the hole). Both can be calculated using the shoelace formula.
How accurate are these calculations?
The formulas are mathematically exact. Accuracy depends on measurement precision of input values. Use more decimal places for higher precision.