Calculating the area of a polygon depends on what kind of polygon you have. There are three reliable methods covering every case: standard formulas for regular polygons, decomposition for composite shapes, and the Shoelace Formula for any irregular polygon defined by vertex coordinates.
A regular polygon has all sides equal and all interior angles equal. Use these direct formulas (s = side length, n = number of sides, a = apothem = perpendicular distance from center to side):
| Shape | Area formula |
|---|---|
| Equilateral triangle (n=3) | A = (√3 / 4) × s² |
| Square (n=4) | A = s² |
| Regular pentagon (n=5) | A ≈ 1.7205 × s² |
| Regular hexagon (n=6) | A = (3√3 / 2) × s² ≈ 2.598 × s² |
| Any regular n-gon | A = ½ × n × s × a |
| Any regular n-gon (apothem unknown) | A = (1/4) × n × s² × cot(π/n) |
Example. Regular hexagon with side 4 cm.
A = (3√3 / 2) × 16 = 24√3 ≈ 41.57 cm².
For polygons made of recognizable sub-shapes (L-shape, T-shape, an arrow, a star), decompose the figure into non-overlapping triangles, rectangles, or trapezoids. Compute each sub-area and sum them. For “negative space” (a hole or cut-out), subtract that area from the bounding figure.
Example. An L-shaped office floor: 12 m × 4 m main area with a 5 m × 3 m extension.
Main = 48, extension = 15, total = 63 m².
This is exactly what our Composite Figure Calculator does — describe the figure or upload a photo, and AI handles the decomposition.
The most powerful method: works for ANY polygon defined by the (x, y) coordinates of its vertices. Even non-convex shapes.
The formula:
A = ½ |Σᵢ (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ)|
List vertices in order (clockwise OR counter-clockwise — the absolute value handles either direction). For each adjacent pair, multiply diagonally, alternate signs. The “shoelace” name comes from the criss-cross visual pattern.
Step-by-step example. Quadrilateral with vertices (0, 0), (4, 0), (5, 3), (1, 4).
| Vertex | x | y | xᵢ·yᵢ₊₁ | xᵢ₊₁·yᵢ |
|---|---|---|---|---|
| A | 0 | 0 | 0×0 = 0 | 4×0 = 0 |
| B | 4 | 0 | 4×3 = 12 | 5×0 = 0 |
| C | 5 | 3 | 5×4 = 20 | 1×3 = 3 |
| D | 1 | 4 | 1×0 = 0 | 0×4 = 0 |
Sum of column 4 = 32. Sum of column 5 = 3. Difference = 29. Area = ½ × |29| = 14.5 sq units.
For regular polygons: Polygon Sides Calculator. For irregular polygons by coordinates: Polygon Coordinates Calculator. For everything else (composites, holes, curved sub-shapes): Composite Figure Calculator.
Does the Shoelace Formula work for non-convex polygons? Yes, as long as the polygon is “simple” (no self-intersecting edges). The formula handles concave shapes (like an arrow or L-shape) without modification.
What’s an apothem? The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It plays the same role as the radius for a circle in the formula A = ½ × perimeter × apothem.
How do I find polygon area in surveying / GPS coordinates? Use the Shoelace Formula directly on the (latitude, longitude) coordinates — but be aware that for large areas you need to project to a planar coordinate system first (UTM, State Plane), since the Shoelace assumes flat geometry.