Hexagon Calculator
Find all measurements of a regular hexagon instantly. Enter the side length and get area, perimeter, and diagonal length in one calculation.
Regular Hexagon Properties and Formulas
A regular hexagon possesses remarkable symmetry. It has six equal sides, six equal angles (each 120 degrees), and six lines of symmetry. You can divide it into six equilateral triangles by drawing lines from the center to each vertex, which makes area calculations straightforward.
Each equilateral triangle has an area of (√3/4) s², where s is the side length. Multiply by six triangles to get the total hexagon area: 6 × (√3/4) s² = (3√3/2) s². Numerically, this is approximately 2.598 times the square of the side length.
The hexagon has two types of diagonals. The long diagonal spans from one vertex straight through the center to the opposite vertex, with length 2s. The short diagonal connects two vertices separated by one intervening vertex, with length s√3 (approximately 1.732s). A regular hexagon has three long diagonals and six short diagonals.
Hexagons in Nature and Design
Honeybees construct hexagonal honeycomb cells because the shape minimizes wax usage while maximizing storage capacity. Mathematically, among all shapes that tile a plane, the regular hexagon has the smallest perimeter for a given area, making it the most material-efficient tessellation.
Basalt rock formations like the Giant's Causeway in Ireland naturally form hexagonal columns when lava cools and contracts. The stress patterns create cracks that meet at 120-degree angles, producing hexagonal cross-sections. Similar patterns appear in dried mud and certain crystal structures.
Engineers use hexagonal grids for everything from tile patterns to structural reinforcement. Hexagonal bolt heads and nuts prevent slipping better than squares. Game designers favor hex grids because movement in six directions is more balanced than the four directions of square grids. This shape's combination of efficiency, symmetry, and practicality makes it ubiquitous in both nature and human design.
Calculating Hexagon Dimensions
When you know one measurement of a regular hexagon, you can find all others using fixed ratios. If you have the side length s, the perimeter is 6s, the long diagonal is 2s, and the area is (3√3/2) s². Conversely, if you know the area A, you can solve for side length: s = √(2A / 3√3).
The radius of the circumscribed circle (passing through all vertices) equals the side length s. The radius of the inscribed circle (touching the midpoint of each side) equals (√3/2) s, which is about 0.866s. These relationships make hexagons particularly easy to construct with a compass—simply walk the compass around a circle six times without changing the radius setting.
In practical applications, hexagon calculations help with tiling floors, designing nuts and bolts, planning gardens, and creating game boards. This calculator handles the somewhat complex area formula (involving √3) automatically, so you can focus on your project rather than manual computation.
Frequently Asked Questions
What is a regular hexagon?
A regular hexagon is a six-sided polygon where all sides have equal length and all interior angles measure 120 degrees. It can be divided into six equilateral triangles meeting at the center.
What is the formula for hexagon area?
For a regular hexagon with side length s, the area is A = (3√3/2) s², which equals approximately 2.598 s². This comes from dividing the hexagon into six equilateral triangles, each with area (√3/4) s².
How do I find the perimeter of a hexagon?
Since a regular hexagon has six equal sides, the perimeter is simply P = 6s, where s is the side length. If each side is 5 cm, the perimeter is 30 cm.
What is the long diagonal of a hexagon?
The long diagonal connects two opposite vertices through the center. For a regular hexagon with side s, the long diagonal equals 2s. There are also short diagonals with length s√3.
Why do hexagons appear in nature?
Hexagons efficiently tile a plane with no gaps, maximizing area while minimizing perimeter. Honeybees build hexagonal cells because this shape uses the least wax while storing the most honey. Hexagons also appear in basalt columns, snowflakes, and insect eyes.