Regular Polygon Calculator
Calculate area, perimeter, interior angles, and apothem of any regular polygon. Enter values for instant results with step-by-step formulas.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Area = (1/2) x Perimeter x Apothem = (n x s^2) / (4 x tan(pi/n))
Where n = number of sides, s = side length. The apothem = s / (2 x tan(pi/n)), circumradius = s / (2 x sin(pi/n)), interior angle = (n-2) x 180 / n, and diagonals = n(n-3)/2.
Worked Examples
Example 1: Regular Hexagon with Side Length 10
Problem:Find the area, perimeter, apothem, and interior angle of a regular hexagon with side length 10 units.
Solution:Interior angle = (6 - 2) x 180 / 6 = 120 degrees\nPerimeter = 6 x 10 = 60 units\nApothem = 10 / (2 x tan(pi/6)) = 10 / (2 x 0.5774) = 8.6603 units\nArea = (1/2) x 60 x 8.6603 = 259.8076 square units\nCircumradius = 10 / (2 x sin(pi/6)) = 10 / 1.0 = 10 units
Result:Area: 259.8076 sq units | Perimeter: 60 units | Apothem: 8.6603 units | Interior Angle: 120 degrees
Example 2: Regular Pentagon with Side Length 7
Problem:Calculate all geometric properties of a regular pentagon with side length 7 units.
Solution:Interior angle = (5 - 2) x 180 / 5 = 108 degrees\nPerimeter = 5 x 7 = 35 units\nApothem = 7 / (2 x tan(pi/5)) = 7 / (2 x 0.7265) = 4.8163 units\nArea = (1/2) x 35 x 4.8163 = 84.2854 square units\nDiagonals = 5 x (5 - 3) / 2 = 5
Result:Area: 84.2854 sq units | Perimeter: 35 units | Apothem: 4.8163 units | Diagonals: 5
Frequently Asked Questions
What is a regular polygon and how is it different from an irregular polygon?
A regular polygon is a closed two-dimensional shape where all sides have equal length and all interior angles are equal in measure. This symmetry makes regular polygons highly predictable and easy to calculate. An irregular polygon, by contrast, can have sides of different lengths and angles of different measures, making calculations much more complex. Common regular polygons include equilateral triangles, squares, regular pentagons, and regular hexagons. Regular polygons are found extensively in nature, architecture, and engineering because their symmetry distributes forces evenly and creates aesthetically pleasing designs.
How do you calculate the interior angle of a regular polygon?
The interior angle of a regular polygon is calculated using the formula (n - 2) times 180 divided by n, where n is the number of sides. This formula works because any polygon can be divided into (n - 2) triangles, and each triangle contains 180 degrees. For example, a regular hexagon with 6 sides has interior angles of (6 - 2) times 180 divided by 6, which equals 120 degrees. As the number of sides increases, the interior angle approaches but never reaches 180 degrees, which is why polygons with many sides begin to look like circles.
What is the apothem and why is it important in polygon calculations?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is critically important because it provides the simplest way to calculate the area of a regular polygon using the formula Area equals one-half times perimeter times apothem. The apothem is also the radius of the largest circle that can be inscribed inside the polygon, known as the incircle. For a regular polygon with side length s and n sides, the apothem equals s divided by 2 times the tangent of pi divided by n. Engineers and architects frequently use the apothem when designing hexagonal tiles, bolt patterns, and other symmetric structures.
How do you find the area of a regular polygon?
The area of a regular polygon can be calculated using the formula A equals one-half times perimeter times apothem. First compute the perimeter by multiplying the number of sides by the side length, then compute the apothem using s divided by 2 times tangent of pi over n. Alternatively, you can use A equals (n times s squared) divided by (4 times tangent of pi over n), which combines both steps into a single formula. For example, a regular hexagon with side length 10 has an area of approximately 259.81 square units. This formula works because the polygon can be divided into n identical isosceles triangles radiating from the center, each with base s and height equal to the apothem.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy