Hexagon Calculator

Calculate the area, perimeter, diagonal, and other properties of a regular hexagon.

Results

Area

64.95

Perimeter

30.00

Long Diagonal

10.00

Visual Comparison

Hexagon Calculator

PropertyValue
Area64.9519
Perimeter30.0000
Long Diagonal10.0000
Circumradius5.0000
Inradius (Apothem)4.3301

About the Regular Hexagon

What Is a Regular Hexagon?

A regular hexagon is a six-sided polygon with all sides equal and all angles equal at 120 degrees each. The hexagon is one of the most efficient shapes in nature, appearing in honeycombs, crystal structures, and basalt columns. It tessellates perfectly, filling a plane with no gaps.

Area Formula

The area of a regular hexagon with side length s is A = (3sqrt(3)/2) x s². This derives from dividing the hexagon into 6 equilateral triangles, each with side s and area (sqrt(3)/4)s².

Special Properties

The circumradius (distance from center to vertex) equals the side length: R = s. The apothem (distance from center to side midpoint) is r = s x sqrt(3)/2. The long diagonal equals 2s (diameter of the circumcircle). These elegant relationships make hexagon calculations particularly straightforward.

Applications

Hexagons are used extensively in engineering and design: honeycomb structures in aerospace, bolt heads and nuts, floor tiles, board game grids, and cellular network design. The hexagonal packing provides maximum area coverage with minimum perimeter.

Practical Example

Regular hexagon with side = 5

Step 1: Area = (3sqrt(3)/2) x 25 = 64.95

Step 2: Perimeter = 6 x 5 = 30

Step 3: Long Diagonal = 2 x 5 = 10

Step 4: Circumradius R = 5

Step 5: Apothem = 5 x sqrt(3)/2 = 4.33

Preguntas Frecuentes

Why does the circumradius equal the side length?

A regular hexagon is composed of 6 equilateral triangles. Since each triangle has all sides equal, the distance from the center to any vertex (circumradius) equals the side length.

How many diagonals does a hexagon have?

A hexagon has 9 diagonals total. The formula is n(n-3)/2 = 6(3)/2 = 9. Three are long diagonals (connecting opposite vertices) and 6 are short diagonals.

What is the apothem?

The apothem is the distance from the center to the midpoint of any side. It equals s x sqrt(3)/2 and is also the inradius of the hexagon.

Why are honeycombs hexagonal?

Hexagons provide the most efficient tiling pattern, using the least material to create a lattice of cells with a given volume. This principle of optimal efficiency is known as the honeycomb conjecture.

Can I calculate properties for irregular hexagons?

This calculator is designed for regular hexagons (equal sides and angles). Irregular hexagons require different methods, typically breaking them into triangles or using coordinate geometry.

Disclaimer: This calculator uses standard geometric formulas. Results are for informational purposes only.

References

  1. Wikipedia. "Regular polygon." en.wikipedia.org
  2. Wolfram MathWorld. mathworld.wolfram.com

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