Regular Polygon Calculator
Compute area, perimeter, and interior angle for any regular polygon. Enter the number of sides and choose from side length, radius, or apothem.
Understanding Regular Polygon Geometry
A regular polygon combines two key properties: all sides are equal in length, and all interior angles are equal. This dual symmetry simplifies calculations dramatically. Any regular polygon can be inscribed in a circle (circumcircle) and can have a circle inscribed within it (incircle).
The circumradius R is the radius of the circumcircle, connecting the center to any vertex. The apothem a is the radius of the incircle, connecting the center to the midpoint of any side. These two radii relate to the side length s through trigonometry: s = 2R sin(ฯ/n) and a = R cos(ฯ/n).
The interior angle formula ((n - 2) ร 180) / n comes from the fact that any n-sided polygon can be divided into (n - 2) triangles, each contributing 180 degrees. Divide the total by n angles to get each individual angle.
Practical Uses of Regular Polygon Calculations
Regular polygons appear in mechanical design, especially in fasteners and gears. Hexagonal bolt heads provide six flat surfaces for wrenches, distributing torque evenly. Knowing the area helps determine material strength and stress limits.
In architecture, octagonal and dodecagonal designs create aesthetically pleasing structures. The symmetry ensures balanced load distribution. Churches, gazebos, and rotundas often feature regular polygon floor plans. Area and perimeter calculations guide material ordering and space planning.
Nature also favors regular polygons. Honeycomb cells are nearly perfect hexagons because that shape tiles efficiently with minimal material. Quasicrystals and certain minerals exhibit pentagonal and decagonal symmetries. Understanding these shapes helps in materials science and crystallography.
From Side Length to Circumradius to Apothem
This calculator accepts three different inputsโside length, circumradius, or apothemโand computes the rest. If you know the side length s, the apothem is a = s / (2 tan(ฯ/n)). The circumradius is R = s / (2 sin(ฯ/n)).
If you have the circumradius R, the side length is s = 2R sin(ฯ/n), and the apothem is a = R cos(ฯ/n). These formulas come from dividing the polygon into n isosceles triangles, each with apex at the center.
If you know the apothem a, the side length is s = 2a tan(ฯ/n), and the circumradius is R = a / cos(ฯ/n). Once you have any one measurement, trigonometry unlocks all the others. This flexibility makes the calculator useful for a wide range of input scenarios.
Frequently Asked Questions
What is the interior angle of a regular polygon?
The interior angle equals ((n - 2) ร 180) / n degrees, where n is the number of sides. For example, a hexagon has interior angles of 120 degrees.
What is the difference between the circumradius and the apothem?
The circumradius is the distance from the center to a vertex. The apothem is the distance from the center to the midpoint of a side. The apothem is always shorter.
How do I find the area from the apothem?
Area equals (perimeter ร apothem) / 2. Multiply the number of sides by the side length to get perimeter, then multiply by the apothem and divide by two.
Can I use this for a triangle or square?
Yes. An equilateral triangle is a regular 3-sided polygon, and a square is a regular 4-sided polygon. This calculator handles all regular polygons.
Why do regular polygons have equal angles?
Regular polygons have all sides equal and all angles equal by definition. The symmetry forces the angles to be identical. The formula ((n - 2) ร 180) / n derives from dividing the total interior angle sum by the number of angles.