Regular Polygon
Compute area, perimeter, apothem, and interior angles of any regular polygon from side count and side length. Free online geometry tool for n-sided shapes.
About This Calculator
A regular polygon is a geometric shape where all sides are equal in length and all interior angles are equal. Common examples include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), regular hexagons (6 sides), and regular octagons (8 sides). This calculator computes all key properties of any regular polygon from just two inputs: the number of sides (n) and the side length (s).
The area is calculated using A = (n x s^2) / (4 x tan(pi/n)) or equivalently A = (perimeter x apothem) / 2. The perimeter is P = n x s. The apothem (incircle radius) is a = s / (2 x tan(pi/n)). The interior angle at each vertex is (n - 2) x 180 deg / n, and the exterior angle is 360 deg / n. The sum of all interior angles always equals (n - 2) x 180 deg. The circumradius (circumscribed circle radius) is R = s / (2 x sin(pi/n)).
Applications: Regular polygons are fundamental in architecture, engineering, design, and mathematics. They appear in building floor plans (octagonal rooms, hexagonal tiles), mechanical parts (hexagonal bolts and nuts), nature (honeycomb cells, basalt columns), and art (Islamic geometric patterns, mosaics). Understanding polygon properties helps in structural design, material estimation, and geometric problem-solving across many disciplines.
Regional Notes: Regular polygon calculations use standard Euclidean geometry formulas and are consistent worldwide. The calculator works with any unit system -- enter side lengths in meters, centimeters, inches, feet, or any other unit, and results will be in the same unit squared for area and the same unit for perimeter, apothem, and circumradius.
Frequently Asked Questions
What is a regular polygon?
A regular polygon is a polygon with all sides equal in length and all interior angles equal. Common examples include equilateral triangles, squares, regular pentagons, and regular hexagons. The formulas for area, perimeter, apothem, and interior angle all depend on the number of sides (n) and the side length (s).
How do you calculate the area of a regular polygon?
The area of a regular polygon is calculated using A = (n x s^2) / (4 x tan(pi/n)), where n is the number of sides and s is the side length. Alternatively, area = (perimeter x apothem) / 2. Our calculator uses both formulas and displays all intermediate values including the apothem, perimeter, and interior angle.
What is the apothem of a regular polygon?
The apothem is the distance from the center of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle. The apothem is calculated as a = s / (2 x tan(pi/n)), where s is the side length and n is the number of sides. The apothem is needed to compute the area as A = (perimeter x apothem) / 2.
How do you find the interior angle of a regular polygon?
Each interior angle of a regular polygon is calculated using the formula (n - 2) x 180 deg / n, where n is the number of sides. For example, a regular hexagon (6 sides) has interior angles of (6-2) x 180 deg / 6 = 120 deg. The sum of all interior angles is (n - 2) x 180 deg.
What regular polygons have the largest area for a given perimeter?
For a fixed perimeter, a circle encloses the maximum area among all shapes. Among regular polygons, as the number of sides increases, the shape becomes closer to a circle and the area increases. That is why a regular icosagon (20 sides) has a larger area than a square with the same perimeter. The limit as n approaches infinity is a circle.
How is the perimeter of a regular polygon calculated?
The perimeter of a regular polygon is simply the number of sides multiplied by the side length: P = n x s. For a regular hexagon with side length 5 units, the perimeter is 6 x 5 = 30 units.
What is the circumradius of a regular polygon?
The circumradius is the distance from the center of a regular polygon to any vertex. It is the radius of the circumscribed circle that passes through all vertices. The circumradius is calculated as R = s / (2 x sin(pi/n)), where s is the side length and n is the number of sides. The circumradius is always larger than the apothem (incircle radius) for any polygon with more than 3 sides.
Is the Regular Polygon Calculator free to use?
Yes, all calculators on Calculy are completely free to use. No registration, login, or payment is required. You can bookmark any calculation with shareable URLs that preserve your inputs.