What is an apothem in a regular polygon?

The apothem is the line segment from the geometric center of a regular polygon perpendicular to the midpoint of one of its sides. It also equals the radius of the inscribed circle (inradius).

What is the sum of interior angles in an n-sided polygon?

The sum of all interior angles is always (n - 2) × 180 degrees. For example, a pentagon has (5 - 2) × 180° = 540°, while an octagon has (8 - 2) × 180° = 1,080°.

How do regular polygons approximate circles?

As the number of sides n approaches infinity, the apothem approaches the circumradius, the perimeter approaches the circle's circumference 2πR, and the polygon's area approaches πR².

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Regular Polygon Calculator

Calculate side length, apothem (inradius), circumradius, area, perimeter, interior angles, and exterior angles for any regular polygon (n = 3 to 100).

Interactive Regular Polygon Calculator Workspace

Live Solver

Regular Polygon Calculation: Solve side length, apothem (inradius), circumradius, area, perimeter, and interior/exterior angles for any regular n-gon from n = 3 to 100.

Number of Sides (n = 6)Regular Hexagon

Known Parameter

Enter Numerical Value

Regular Hexagon Outputs

Area (A)64.9519

Perimeter (P)30.0000

Apothem (a)4.3301

Circumradius (R)5.0000

Interior Angle120.00°

Exterior Angle60.00°

Apothem & Trigonometric Breakdown

1. Central Angle (2α): 360° / 6 = 60.00° (half-angle α = 30.00°)

2. Apothem formula: a = s / [2 × tan(180°/6)] = 4.3301

3. Area formula: Sum of 6 isosceles central triangles: Area = (1/2) × Perimeter × Apothem = 0.5 × 30.0000 × 4.3301 = 64.9519

4. Sum of Interior Angles: (6 - 2) × 180° = 720°

Mathematical Formula & Variables

Equation ModelArea = (1/2) × n × s × a = (1/2) × P × a, Apothem a = s / [2 × tan(180°/n)]

Computes geometric properties of regular equiangular and equilateral n-gons using trigonometric ratios and triangular decomposition.

Variable Definitions

Symbol	Description
n (Number of Sides)	Total count of equal straight boundary edges (n ≥ 3).
s (Side Length)	Linear measure of one outer boundary edge.
a (Apothem / Inradius)	Perpendicular distance from the polygon center to the midpoint of any side.
R (Circumradius)	Distance from the center point to any outer vertex.
P (Perimeter)	Total boundary line length: P = n × s.

How to Use the Regular Polygon Calculator

Select or enter the number of sides (n) for your regular polygon (e.g. 5 for pentagon, 6 for hexagon, 8 for octagon).
Select which parameter you already know: Side Length, Apothem, Circumradius, or Area.
Type your numerical measurement into the input box.
Instantly view the calculated Area, Perimeter, Apothem, Circumradius, Interior Angle, and Exterior Angle.

Practical Example Calculation

Scenario Context: A carpenter is crafting a regular hexagonal tabletop (n = 6) with each outer edge measuring 40 cm.

Step 1: Identify given values: Number of sides (n) = 6, Side length (s) = 40 cm.

Step 2: Calculate Perimeter (P): P = n × s = 6 × 40 = 240 cm.

Step 3: Calculate Central Half-Angle: α = 180° / n = 180° / 6 = 30°.

Step 4: Calculate Apothem (a): a = s / [2 × tan(30°)] = 40 / [2 × 0.57735] ≈ 34.64 cm.

Step 5: Calculate Area (A): A = 0.5 × P × a = 0.5 × 240 × 34.641 = 4,156.92 square centimeters.

Step 6: Calculate Interior Angle: ((6 - 2) × 180°) / 6 = 720° / 6 = 120°.

The regular hexagonal tabletop has an area of 4,156.92 cm², perimeter of 240 cm, apothem of 34.64 cm, and interior angles of 120°.

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Regular Polygons Explained: Apothem, Interior Angles & Area FormulasAcademic guide & formulas