Use this Polygon Calculator to find the area, perimeter, and interior angles of a regular polygon. Enter the number of sides and a known measurement, such as the side length or apothem, to calculate the remaining values. You can also change the measurement unit to suit your needs.
Polygon Details
Enter a whole number from 3 to 1,000.
Enter the length of one side.
Polygon Results
Regular polygon preview
This calculator assumes a regular polygon, meaning all sides and all interior angles are equal.
What Is a Polygon?
A polygon is a flat, closed shape made up of straight sides. Triangles, squares, pentagons, hexagons, and octagons are all examples of polygons.
A regular polygon has sides of equal length and equal interior angles. This Polygon Calculator helps you find the area, perimeter, apothem, interior angle, and exterior angle of a regular polygon.
Simply enter the number of sides and provide either the side length or the apothem. The calculator will work out the remaining measurements automatically.
Polygon Formulas
You can use the following formulas to calculate the main properties of a regular polygon.
1. Perimeter
The perimeter is the total distance around the polygon.
Perimeter = Number of sides × Side length (n x a)
2. Area
When the side length is known:
Area = (n × s²) ÷ (4 × tan(180° ÷ n))
When the apothem is known:
Area = (Perimeter × Apothem) ÷ 2
3. Apothem
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides.
Apothem = Side length ÷ (2 × tan(180° ÷ n))
4. Interior Angle (α)
An interior angle is the angle inside the polygon at each vertex.
α = (n − 2) × 180° ÷ n
In radians, the formula is:
α = (n − 2) × π ÷ n
5. Exterior Angle (β)
An exterior angle is formed by extending one side of the polygon. For a regular polygon, all exterior angles are equal.
β = 360° ÷ n
In radians, the formula is:
β = 2 × π ÷ n
6. Sum of Interior Angles
To find the total of all interior angles, use:
Sum of interior angles = (n − 2) × 180°
In these formulas, n represents the number of sides and s represents the side length.
Polygon Calculator Examples
Example 1: Regular Hexagon
Suppose you have a regular hexagon with six sides, each measuring 5 cm.
- Number of sides: 6
- Side length: 5 cm
- Perimeter: 6 × 5 = 30 cm
- Apothem: approximately 4.33 cm
- Area: approximately 64.95 cm²
- Interior angle: 120°
- Exterior angle: 60°
- Sum of interior angles: 720°
Example 2: Regular Pentagon
Suppose a regular pentagon has five equal sides, each measuring 4 cm.
- Number of sides: 5
- Side length: 4 cm
- Perimeter: 5 × 4 = 20 cm
- Apothem: approximately 2.75 cm
- Area: approximately 27.53 cm²
- Interior angle: 108°
- Exterior angle: 72°
- Sum of interior angles: 540°
Example 3: Regular Octagon
Consider a regular octagon with eight sides, each measuring 3 cm.
- Number of sides: 8
- Side length: 3 cm
- Perimeter: 8 × 3 = 24 cm
- Apothem: approximately 3.62 cm
- Area: approximately 43.46 cm²
- Interior angle: 135°
- Exterior angle: 45°
- Sum of interior angles: 1,080°
Example 4: Calculate a Hexagon Using Its Apothem
Suppose a regular hexagon has an apothem of 6 cm.
- Number of sides: 6
- Apothem: 6 cm
- Side length: approximately 6.93 cm
- Perimeter: approximately 41.57 cm
- Area: approximately 124.71 cm²
- Interior angle: 120°
- Exterior angle: 60°
To check this example, select Apothem as the known measurement, enter 6, and choose centimeters from the unit selector beside the input.
How to Use the Polygon Calculator
- Enter the number of sides. For example, enter 5 for a pentagon or 8 for an octagon.
- Choose the measurement you already know: side length or apothem.
- Enter its value in the measurement field.
- Select the appropriate unit from the dropdown beside the measurement field. Available units include millimeters, centimeters, meters, inches, feet, and yards.
- Read the results to find the area, perimeter, apothem, interior angle, exterior angle, and sum of interior angles.
The calculator updates the results automatically whenever you change an input.
Frequently Asked Questions
How do I calculate the area of a regular polygon?
Multiply the perimeter by the apothem and divide the result by two. If you know the side length and number of sides instead, you can use the regular polygon area formula to calculate the area directly.
What is the difference between an interior angle and an exterior angle?
An interior angle lies inside the polygon at a vertex. An exterior angle is formed between one side and the extension of an adjacent side. For a regular polygon, the interior and exterior angles at each vertex add up to 180°.
What is the sum of a polygon's interior angles?
The sum is calculated using (n − 2) × 180°, where n is the number of sides. For example, a hexagon has six sides, so its interior angles add up to 720°.
What is the sum of a regular polygon's exterior angles?
The sum of one exterior angle at each vertex of a convex polygon is always 360°. For a regular polygon, each exterior angle equals 360° divided by the number of sides.
What is an apothem in geometry?
An apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides. It is useful when calculating the polygon's area.
Can I calculate the area of an irregular polygon?
This calculator is designed for regular polygons, where all sides and interior angles are equal. An irregular polygon generally requires additional measurements, such as its vertices or individual side lengths and angles.
Does the calculator support different measurement units?
Yes. You can select millimeters, centimeters, meters, inches, feet, or yards using the dropdown beside the measurement field. Lengths appear in the selected unit, while areas appear in the corresponding square unit.
What is the difference between a polygon and a regular polygon?
A polygon is any closed, two-dimensional shape formed by straight sides. A regular polygon has all sides equal in length and all interior angles equal.
How many sides can I enter in the calculator?
You can enter any whole number from 3 to 1,000. Three sides form a triangle, five form a pentagon, six form a hexagon, and eight form an octagon.
Note: The calculator assumes a regular polygon. Results are rounded for display, so small differences may occur when comparing them with calculations using more decimal places.