Regular Polygon Apothem Calculator
The apothem of a regular polygon is the perpendicular distance from its center to the midpoint of any side, equal to the radius of the inscribed circle. It is the key dimension for finding a polygon's area, since the area equals one half the apothem times the perimeter. This calculator finds the apothem from two numbers: the length of one side and the number of sides. It uses the standard trigonometric relationship that the apothem equals the side length divided by twice the tangent of pi over the number of sides. The angle pi over n is half the central angle of one side, expressed in radians, and the tangent ties that angle to the ratio of half the side to the apothem. The formula works for any regular polygon, from an equilateral triangle through squares, pentagons and hexagons to many-sided shapes that approximate a circle. Architects, tilers, machinists, drafters and students use the apothem to lay out regular shapes and compute areas. Enter your side length and number of sides to get the apothem immediately, with both inputs editable. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A regular polygon's apothem is s / (2 tan(pi / n)). For side s = 6 and n = 6 sides, the apothem is 5.20.
Regular Polygon Apothem formula
a = s / (2 tan(pi / n))
a = apothem (center to side midpoint)
s = side length
n = number of sides
pi / n is in radians
The angle pi over n is half the central angle subtended by one side. Its tangent equals half the side length divided by the apothem, so rearranging gives the apothem directly.
Worked example
Find the apothem of a regular hexagon with side length 6 (six sides).
- Angle = pi / 6 = 0.523599 radians
- tan(pi / 6) = 0.577350
- a = 6 / (2 x 0.577350) = 6 / 1.154701 = 5.20
The apothem is 5.20. These are the calculator's default inputs, so the result above matches the widget exactly.
Apothem for side length 6 at common side counts
a = 6 / (2 tan(pi / n)).
| Sides (n) | Apothem |
|---|---|
| 3 | 1.73 |
| 4 | 3.00 |
| 5 | 4.13 |
| 6 | 5.20 |
| 8 | 7.24 |
Mathematical functions reference: US National Institute of Standards and Technology (NIST).
Regular Polygon Apothem Calculator: frequently asked questions
What is the apothem of a polygon?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides. It equals the radius of the largest circle that fits inside the polygon, the inscribed circle. It is shorter than the distance from the center to a vertex, which is the circumradius.
Why is the apothem useful?
It gives a quick way to find the area of a regular polygon, which equals one half the apothem times the perimeter. It also sets out the inscribed circle and helps machinists and drafters position features evenly inside a regular shape.
Does the formula need radians?
Yes. The term pi over n is an angle in radians, equal to half the central angle of one side. If your calculator works in degrees, use 180 over n degrees instead, but the underlying tangent relationship is the same. This tool computes the tangent in radians internally.
What is the smallest number of sides allowed?
Three, an equilateral triangle. A polygon needs at least three sides to enclose an area. As the number of sides grows, the apothem approaches the circumradius and the polygon looks more and more like a circle.
What is the apothem formula?
The apothem equals the side length divided by twice the tangent of pi over the number of sides: a = s / (2 tan(pi / n)). For a hexagon with side 6 that is 6 divided by 1.154701, about 5.20.
Official sources
- Mathematical functions and computational reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.