Calculate Polygon Area
Choose a calculation method. Regular polygons have equal sides and equal angles. For an irregular polygon, enter the coordinates of its vertices in order around the boundary.
A regular polygon can be calculated from its side length, apothem, or circumradius. The selected measurement uses the unit chosen above.
Polygon preview
The diagram is a visual guide and is not intended as a dimensionally accurate engineering drawing.
Polygon Area Formulas
1. Regular polygon from side length
A regular polygon has the same side length and interior angle at every side and vertex. If it has n sides of length s, its area can be calculated directly.
The perimeter is P = n × s, and the apothem is a = s / [2 × tan(π / n)].
2. Regular polygon from apothem
Here, P is the perimeter and a is the apothem, the perpendicular distance from the center to the midpoint of a side. When the apothem is known, the side length is s = 2a × tan(π / n).
3. Regular polygon from circumradius
The circumradius R is the distance from the polygon's center to any vertex. The side length is s = 2R × sin(π / n).
4. Irregular polygon: shoelace formula
For an irregular polygon with ordered vertices (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ), multiply each vertex's x-coordinate by the next vertex's y-coordinate, subtract the opposite products, add the results, and take half the absolute value.
The final vertex connects back to the first vertex. This method works for simple polygons whose vertices are listed around the boundary.
Further reading: Regular polygon area formulas and The polygon area formula.
Worked Polygon Area Examples
Example 1: Regular hexagon with side length 5 cm
A hexagon has six sides. Its area is 6 × 5² / [4 × tan(π/6)]. The result is approximately 64.9519 cm². Its perimeter is 6 × 5 = 30 cm.
Example 2: Square with side length 4 m
A square is a regular polygon with four sides. Its area is 4 × 4² / [4 × tan(π/4)] = 16 m².
Example 3: Regular pentagon with side length 3 m
The area is 5 × 3² / [4 × tan(π/5)], approximately 15.484 m².
Example 4: Irregular polygon from coordinates
Consider the vertices (0,0), (6,0), (7,4), (3,6), and (0,3). Applying the shoelace formula gives an area of 31.5 square units.
The coordinate example's result uses the same unit for both coordinates; its area is therefore expressed in that unit squared.
Common Polygon Types
A polygon is a closed two-dimensional shape formed by straight line segments. Its name usually depends on its number of sides.
| Sides | Name | Regular polygon area formula |
|---|---|---|
| 3 | Triangle | A = 3s² / [4 tan(π/3)] |
| 4 | Square | A = s² |
| 5 | Pentagon | A = 5s² / [4 tan(π/5)] |
| 6 | Hexagon | A = 3√3s² / 2 |
| 7 | Heptagon | A = 7s² / [4 tan(π/7)] |
| 8 | Octagon | A = 2(1 + √2)s² |
| 10 | Decagon | A = 10s² / [4 tan(π/10)] |
Regular vs. irregular polygons
A regular polygon has equal sides and equal interior angles. An irregular polygon does not satisfy both conditions. The side-length formula above is for regular polygons only; irregular polygons generally need coordinates or other geometric information to determine their area.
What is the apothem?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one side. It is useful because the polygon can be split into equal triangles, each with a height equal to the apothem.
Frequently Asked Questions
How do you calculate the area of a regular polygon?
Multiply half the perimeter by the apothem. If only the number of sides and side length are known, use A = ns² / [4 tan(π/n)].
How do you find the area of an irregular polygon?
If its vertices are known, list them in order and use the shoelace formula. If coordinates are not available, additional information such as side lengths, angles, or a way to divide the shape into known shapes may be required.
What is the apothem of a polygon?
It is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides. The apothem is not generally defined for an irregular polygon.
Can this calculator calculate a hexagon or octagon?
Yes. Enter 6 for a hexagon or 8 for an octagon and supply the required measurement. You can enter other side counts as well.
Do coordinates need to be in a particular order?
Yes. Enter vertices consecutively around the boundary, clockwise or counterclockwise. Random ordering can create an incorrect boundary and produce an incorrect area.
What units does the polygon area calculator use?
Select a length unit for the regular polygon or coordinate inputs. The area is returned in the corresponding squared unit. For example, measurements in meters produce an area in square meters.
Can a polygon have more than eight sides?
Yes. A polygon can have many sides. This calculator supports regular polygons with up to 1,000 sides, subject to ordinary floating-point calculation limits.
Does the shoelace formula work for self-intersecting polygons?
The standard formula calculates signed area. For self-intersecting shapes, that result may represent a net signed area rather than the total area of all regions. This calculator is intended for simple, non-self-intersecting polygons.