Hexagon Calculator
Calculate area, perimeter, inradius, circumradius, and diagonals of a regular hexagon
Length of one side of the regular hexagon
Total area of the hexagon in square units
Sum of all six sides
Hexagon Properties
Side Length
Perimeter
Area
Inradius (Apothem)
r = (s√3)/2
Circumradius
R = s
Short Diagonal
d₁ = s√3
Long Diagonal
d₂ = 2s
Interior Angle
°
Each interior angle
Exterior Angle
°
Each exterior angle
Hexagon Formulas
Area
A = (3√3/2) × s²
Perimeter
P = 6s
Inradius (Apothem)
r = (s√3)/2
Circumradius
R = s
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About 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 equal (120° each). It is one of the three regular polygons that can tile a plane without gaps (along with equilateral triangles and squares).
Hexagons appear throughout nature—from honeycomb cells to the basalt columns of Giant's Causeway, and even the famous hexagonal storm on Saturn's north pole.
How to Use This Calculator
From Side Length
- Select From Side Length mode
- Enter the side length of your hexagon
- Instantly view all calculated properties: area, perimeter, inradius, circumradius, and diagonals
From Area
- Select From Area mode
- Enter the known area of the hexagon
- The calculator determines the side length and all other properties
From Perimeter
- Select From Perimeter mode
- Enter the total perimeter
- View the calculated side length and all derived properties
Hexagon Formulas
Core Formulas (Regular Hexagon)
| Property | Formula | Description |
|---|---|---|
| Area | A = (3√3/2) × s² | Total area enclosed |
| Perimeter | P = 6s | Sum of all six sides |
| Inradius (Apothem) | r = (s√3)/2 | Distance from center to middle of a side |
| Circumradius | R = s | Distance from center to a vertex |
Diagonal Formulas
| Property | Formula | Description |
|---|---|---|
| Short Diagonal | d₁ = s√3 | Connects vertices with one vertex between |
| Long Diagonal | d₂ = 2s | Connects opposite vertices (passes through center) |
Derived Formulas
| Property | Formula |
|---|---|
| Side from Area | s = √(2A / 3√3) |
| Side from Perimeter | s = P / 6 |
Understanding Hexagon Properties
Interior Angles
Every interior angle of a regular hexagon measures exactly 120°. The sum of all interior angles is (6-2) × 180° = 720°.
Inradius vs Circumradius
- Inradius (apothem): The perpendicular distance from the center to the midpoint of any side. It's the radius of the largest circle that fits inside the hexagon.
- Circumradius: The distance from the center to any vertex. It's the radius of the smallest circle that completely contains the hexagon. For a regular hexagon, R = s (circumradius equals side length).
Diagonals
A regular hexagon has 9 diagonals total:
- 6 short diagonals (length s√3)
- 3 long diagonals (length 2s) that pass through the center
Real-World Applications
Nature & Biology
- Honeycomb cells: Bees use hexagonal cells because they provide the most storage with the least wax
- Turtle shells: Many have hexagonal patterns
- Snowflakes: Form hexagonal crystal structures
Engineering & Architecture
- Nuts and bolts: Hexagonal heads allow for easy gripping with wrenches
- Tiles and flooring: Hexagonal tiles create interesting patterns
- Structural design: Hexagonal grids provide excellent strength-to-weight ratios
Science
- Carbon structures: Graphene and carbon nanotubes are based on hexagonal arrangements
- Saturn's hexagon: A persistent hexagonal cloud pattern at Saturn's north pole
Frequently Asked Questions
Why are hexagons so common in nature?
Hexagons are efficient. They pack together without gaps, minimize perimeter for a given area, and distribute forces evenly. When circular objects are pressed together (like bubbles), they naturally form hexagonal patterns.
What's the difference between a regular and irregular hexagon?
A regular hexagon has all sides equal and all angles equal (120°). An irregular hexagon has sides and/or angles of different measures. This calculator works only for regular hexagons.
How do I measure a hexagon in real life?
Measure one side length, then use this calculator. For physical hexagons, you can also measure the distance between opposite sides (which equals 2 × inradius) or opposite vertices (which equals 2 × circumradius = 2s).
Why does circumradius equal side length for a hexagon?
This is a unique property of regular hexagons. A regular hexagon can be divided into 6 equilateral triangles, each with the same side length as the hexagon. The circumradius is the side length of these triangles, which equals the hexagon's side length.