The Polygon Area Calculator calculates the area of shapes with straight sides — from a regular hexagon or octagon to any irregular polygon with as many corners as you need.
Regular polygons use A = n × s² ÷ (4 × tan(π ÷ n)) from the side length, or A = ½ × n × R² × sin(2π ÷ n) from the center-to-corner radius. Irregular polygons are calculated from their corner coordinates with the shoelace formula, and a quadrilateral mode handles four sides plus one diagonal.
Polygon Area Calculator
Area of regular polygons (hexagon, octagon, any n sides) from side or circumradius, of quadrilaterals, and of irregular polygons from coordinates.
Sides a and b meet the diagonal at one end, sides c and d at the other.
Formula
Regular polygon from side length:
A = n × s² ÷ (4 × tan(π ÷ n)), apothem a = s ÷ (2 tan(π ÷ n))
Regular polygon from circumradius:
A = ½ × n × R² × sin(2π ÷ n), side s = 2R sin(π ÷ n)
Irregular polygon (shoelace formula):
A = ½ |Σ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|
Where: n = number of sides, s = side length, R = circumradius (center to a corner), (xᵢ, yᵢ) = corner coordinates in order.
How the Calculation Works
Step 1: A regular polygon is split into n identical triangles meeting at the center.
Step 2: Each triangle has area ½ × side × apothem, so the total is ½ × perimeter × apothem.
Step 3: For irregular polygons, the shoelace formula adds signed triangle areas between consecutive corners and halves the absolute total.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| n | Number of sides (3 or more) | — |
| s | Side length | cm, m, in, ft |
| R | Circumradius | cm, m, in, ft |
| Apothem | Center to the middle of a side | cm, m, in, ft |
| A | Area | cm², m², in², ft² |
Example Calculation
Regular hexagon
Given: n = 6, side = 4 cm
Calculation: A = 6 × 16 ÷ (4 × tan 30°) = 96 ÷ 2.3094
Result: A ≈ 41.57 cm² (apothem ≈ 3.46 cm)
Irregular hexagon from coordinates
Given: (0, 0), (8, 0), (10, 4), (8, 8), (2, 9), (−1, 4) cm
Result: A = 76.5 cm²; perimeter ≈ 32.98 cm
Regular Polygon Area Table
Multiply the side length squared by the factor to get the area.
| Polygon | Sides | Area factor (× s²) | Interior angle |
|---|---|---|---|
| Equilateral triangle | 3 | 0.4330 | 60° |
| Square | 4 | 1.0000 | 90° |
| Pentagon | 5 | 1.7205 | 108° |
| Hexagon | 6 | 2.5981 | 120° |
| Heptagon | 7 | 3.6339 | ≈128.57° |
| Octagon | 8 | 4.8284 | 135° |
| Nonagon | 9 | 6.1818 | 140° |
| Decagon | 10 | 7.6942 | 144° |
| Dodecagon | 12 | 11.1962 | 150° |
Shape Properties
A polygon is a closed flat shape with straight sides. Its interior angles add up to (n − 2) × 180°. In a regular polygon all sides and angles are equal, and it fits exactly inside a circle of radius R.
Dimensions Explained
Circumradius vs apothem
The circumradius goes from the center to a corner; the apothem goes from the center to the middle of a side and is always shorter. Use the circumradius mode if you know the size of the circle the polygon fits in.
Coordinates order
Enter corners in order around the shape. Sides must not cross — the calculator warns you if they do.
Frequently Asked Questions
How do you find the area of a polygon?
For a regular polygon, use ½ × perimeter × apothem. For an irregular one, use corner coordinates with the shoelace formula or split it into triangles.
How do you find the area of a hexagon?
For a regular hexagon, A = 2.5981 × side².
How do you find the area of an octagon?
For a regular octagon, A = 2(1 + √2) × side² ≈ 4.8284 × side².
How do you find the area of an irregular polygon?
List the corner coordinates in order and apply the shoelace formula.
What is the apothem of a polygon?
The distance from the center to the midpoint of a side: s ÷ (2 tan(π ÷ n)).
