Hexagon Calculator

What is a Regular Hexagon?

A regular hexagon is a six-sided polygon where all sides are equal in length and all interior angles are 120 degrees. It is one of the most efficient shapes in nature and engineering because it can tile the plane perfectly without gaps — the reason honeycomb cells, soccer-ball panels, and Allen-key heads use hexagonal geometry. A regular hexagon decomposes into six equilateral triangles meeting at the center, which gives it the unique property that the circumradius (center-to-vertex distance) equals the side length.

The hexagon is the most area-efficient regular polygon that tiles the plane, which is why bees use it for honeycomb — it provides the maximum storage area for the minimum wall material. In engineering, hexagonal cross-sections appear in nuts, bolts, and wrenches because the six flat sides provide multiple gripping angles.

Formulas and Calculations

The key formulas for a regular hexagon are based on the side length (s):

Area (A): A = (3√3 / 2) × s²

The area formula comes from decomposing the hexagon into six equilateral triangles. Each triangle has area (√3/4)s², so the total is 6 × (√3/4)s² = (3√3/2)s². The coefficient (3√3/2) is approximately 2.598.

Perimeter (P): P = 6s

The perimeter is simply six times the side length.

Apothem (a): a = s√3 / 2

The apothem is the perpendicular distance from the center to the midpoint of any side. It is also the radius of the inscribed circle.

Circumradius (R): R = s

For a regular hexagon only, the circumradius equals the side length — a special property that does not hold for other regular polygons.

Diagonals

Long Diagonal: = 2s (vertex to opposite vertex)

Short Diagonal: = s√3 (edge midpoint to opposite edge midpoint)

Reverse Calculations

Given Area: s = √(2A / 3√3)

Given Perimeter: s = P / 6

Given Apothem: s = 2a / √3

How to Use the Hexagon Calculator

Using this calculator is simple. First, select what value you know from the dropdown menu: side length, area, perimeter, or apothem. Then enter the known value. The calculator will instantly compute all other properties of the regular hexagon in real time, including the area, perimeter, apothem, circumradius, and both diagonals.

All calculations are performed entirely in your browser with no server requests, ensuring instant feedback and privacy. You can enter whole numbers or decimals for precise calculations.

Frequently Asked Questions

How do you calculate the area of a regular hexagon?

Use A = (3√3 / 2) × s², where s is the side length. For example, a hexagon with s = 4 cm has area A = (3√3/2) × 16 = 24√3 ≈ 41.57 cm².

Why does a regular hexagon's circumradius equal its side length?

A regular hexagon decomposes into 6 equilateral triangles meeting at the center. Each triangle has the side length on its base and the circumradius on its two other sides. Since all sides of an equilateral triangle are equal, the circumradius equals the side length. This is unique to hexagons among regular polygons.

What is the apothem of a regular hexagon?

The apothem is the perpendicular distance from the center to the midpoint of any side. For a regular hexagon, apothem = s√3/2. It is also the radius of the inscribed circle that touches each side at its midpoint.

What are the diagonals of a regular hexagon?

A regular hexagon has two distinct diagonals: the long diagonal (vertex to opposite vertex) is 2s, and the short diagonal (edge midpoint to opposite edge midpoint) is s√3. For s = 4 cm, the long diagonal is 8 cm and the short diagonal is approximately 6.928 cm.

Why do hexagons appear so often in nature and engineering?

Hexagons are the most area-efficient regular polygon that can tile the plane without gaps. This means they enclose the maximum area for the minimum perimeter when used in a tiled pattern. This efficiency is why bees use hexagonal honeycomb cells, and why hexagonal tiles, nuts, and bolts are so common in engineering.

What is the difference between a regular hexagon and an irregular hexagon?

A regular hexagon has six equal sides and six equal angles (each 120 degrees). An irregular hexagon has sides and/or angles of different lengths and measures. This calculator handles only the regular case, where all properties follow from a single dimension — the side length.

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