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 for accuracy by Manoj Kumar, Mathematics Educator
Regular Polygon Calculator
Calculator
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Formula: Area = (1/2) x Perimeter x Apothem = (n x s^2) / (4 x tan(pi/n))
Worked example — Area: 259.8076 sq units | Perimeter: 60 units | Apothem: 8.6603 units | Interior Angle: 120 degrees
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 Perimeter = 6 x 10 = 60 units Apothem = 10 / (2 x tan(pi/6)) = 10 / (2 x 0.5774) = 8.6603 units Area = (1/2) x 60 x 8.6603 = 259.8076 square units Circumradius = 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 Perimeter = 5 x 7 = 35 units Apothem = 7 / (2 x tan(pi/5)) = 7 / (2 x 0.7265) = 4.8163 units Area = (1/2) x 35 x 4.8163 = 84.2854 square units Diagonals = 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.
What is the circumradius versus the inradius of a regular polygon?
The circumradius (also called simply the radius) is the distance from the center of the polygon to any vertex, while the inradius is the distance from the center to the midpoint of any side, which is the same as the apothem. The circumradius defines the circumscribed circle that passes through all vertices, while the inradius defines the inscribed circle that is tangent to all sides. The circumradius is always larger than the inradius. For a regular polygon with side length s and n sides, the circumradius equals s divided by 2 times sine of pi over n, and the inradius equals s divided by 2 times tangent of pi over n. The ratio between these two values depends on the number of sides.
How many diagonals does a regular polygon have?
The number of diagonals in any polygon with n sides is given by the formula n times (n minus 3) divided by 2. A triangle has zero diagonals since no vertex can connect to a non-adjacent vertex. A square has 2 diagonals, a pentagon has 5, a hexagon has 9, and a decagon has 35. The formula works because each vertex can connect to n minus 3 other vertices (excluding itself and its two adjacent vertices), giving n times (n minus 3) connections, then dividing by 2 to avoid counting each diagonal twice. In regular polygons, diagonals have interesting symmetry properties and create smaller regular polygons at their intersections.
Why do regular hexagons appear so often in nature and engineering?
Regular hexagons are ubiquitous in nature and engineering because they are the most efficient shape for covering a plane without gaps while minimizing the total perimeter, a property proven mathematically as the honeycomb conjecture. Honeybees use hexagonal cells because this shape holds the most honey with the least wax. In engineering, hexagonal patterns appear in nuts and bolts, floor tiles, and carbon nanostructures like graphene. The 120-degree interior angle of a hexagon is significant because exactly three hexagons meet at each vertex, creating a perfect tessellation. Hexagonal grids also provide more uniform coverage than square grids, which is why they are used in cellular network planning.
Can a regular polygon tessellate a plane?
Only three regular polygons can tessellate the plane by themselves: equilateral triangles, squares, and regular hexagons. For tessellation to work, the interior angles meeting at each vertex must sum to exactly 360 degrees. Equilateral triangles have 60-degree angles, and six of them meet at a point to give 360 degrees. Squares have 90-degree angles, with four meeting at a point. Hexagons have 120-degree angles, with three meeting at a point. Regular pentagons with 108-degree angles cannot tessellate because 360 is not evenly divisible by 108. However, combinations of different regular polygons can create semi-regular tessellations, such as the classic octagon-and-square tile pattern.
How does the shape approach a circle as the number of sides increases?
As the number of sides of a regular polygon increases, both its interior angles and its overall shape approach those of a circle. A regular polygon with 100 sides has interior angles of 176.4 degrees and is virtually indistinguishable from a circle to the naked eye. Mathematically, the area of a regular n-gon with circumradius r approaches pi times r squared as n approaches infinity. Archimedes famously used this principle to approximate pi by computing the perimeters of inscribed and circumscribed 96-sided polygons. This convergence is why engineers often approximate circles with high-sided regular polygons in computer graphics and finite element analysis.
What are real-world applications of regular polygon calculations?
Regular polygon calculations are used extensively across many fields. Architects use them to design floor plans, window shapes, and structural columns. Engineers use octagonal cross-sections for stop signs and hexagonal bolt heads for wrenches. Urban planners use hexagonal grids for cellular tower placement to maximize coverage with minimal overlap. Game designers use hexagonal grids for strategy games because hexagons provide six equidistant neighbors compared to four for squares. In crystallography, regular polygon geometry describes the cross-sections of mineral crystals. Even in everyday life, regular polygons appear in soccer balls, which are composed of regular pentagons and hexagons arranged in a truncated icosahedron pattern.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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