Regular Hexagon Calculator

Calculate area, perimeter, apothem, and diagonals of a regular hexagon from its side length.

Enter Hexagon Side Length

Result

Area
259.807621
square units
Side Length 10 units
Perimeter 60 units
Apothem 8.660254 units
Short Diagonal (d) 17.320508 units
Long Diagonal (D) 20 units
Circumradius (R) 10 units
Interior Angle 120°

Step-by-Step Solution

A = (3sqrt(3)/2) x s^2

What Is a Regular Hexagon?

A regular hexagon is a polygon with six equal sides and six equal angles. Each interior angle measures exactly 120 degrees, and the sum of all interior angles is 720 degrees. The regular hexagon is one of only three regular polygons that can tile the plane (along with the equilateral triangle and square).

Hexagons are ubiquitous in nature. Honeybee combs, the Giant's Causeway basalt columns, and the cloud pattern at Saturn's north pole all feature hexagonal geometry. This prevalence stems from the hexagon's efficiency: it provides the maximum area with the minimum perimeter for a tiling shape.

Hexagon Formulas

Area

The area of a regular hexagon with side length s.

A = (3sqrt(3)/2) x s2

Perimeter

The total length of all six sides.

P = 6s

Apothem

Distance from the center to the midpoint of a side.

a = s x sqrt(3) / 2

Short Diagonal

Connects two vertices separated by one vertex.

d = s x sqrt(3)

Long Diagonal

Connects two opposite vertices (passes through center).

D = 2s

Circumradius

Distance from center to any vertex. Equals the side length.

R = s

Key Properties

  • Sides: 6 equal sides
  • Interior angle: 120 degrees each
  • Exterior angle: 60 degrees each
  • Diagonals: 9 total (3 long diagonals through center, 6 short diagonals)
  • Symmetry: 6 lines of symmetry, rotational symmetry of order 6
  • Composed of: 6 equilateral triangles

Why Hexagons Are Special

A regular hexagon can be decomposed into exactly 6 equilateral triangles, each with side length equal to the hexagon's side. This makes calculations elegant. Also, the circumradius of a regular hexagon equals its side length, a unique property among regular polygons.

Hexagons in Nature and Engineering

The hexagonal packing arrangement maximizes the use of space, which is why bees build hexagonal cells. In engineering, hexagonal bolt heads and nuts are standard because a wrench can grip them at multiple angles (every 60 degrees). Carbon atoms in graphene form a hexagonal lattice, giving it extraordinary strength.

Step-by-Step Example

Find all properties of a regular hexagon with side length s = 10:

  1. Area: A = (3sqrt(3)/2) x 10^2 = (3 x 1.7321 / 2) x 100 = 2.5981 x 100 = 259.81
  2. Perimeter: P = 6 x 10 = 60
  3. Apothem: a = 10 x sqrt(3) / 2 = 10 x 0.8660 = 8.6603
  4. Short diagonal: d = 10 x sqrt(3) = 17.3205
  5. Long diagonal: D = 2 x 10 = 20