Last updated: July 31, 2026
Hexagon Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
A = (3√3/2) × a²
Where:
- A=Area(square units)
- a=Side length(units)
- P=Perimeter(units)
- r=Apothem(units)
- R=Circumradius(units)
- dₗ=Long diagonal(units)
- dₛ=Short diagonal(units)
Worked Examples
Unit hexagon
Reference case for the regular hexagon constants.
- 1A = (3√3/2) × 1² ≈ 2.598076
- 2P = 6 × 1 = 6
- 3Apothem = √3/2 ≈ 0.866025
- 4Circumradius = 1
Hexagon with side 4
Useful for checking apothem and diagonal formulas.
- 1A = (3√3/2) × 16 ≈ 41.569219
- 2Short diagonal = 4√3 ≈ 6.928203
- 3Long diagonal = 8
Hexagon with side 7
Demonstrates that circumradius equals side length.
- 1Circumradius = a = 7
- 2Perimeter = 6 × 7 = 42
- 3Long diagonal = 2 × 7 = 14
Invalid: side = 0
A regular hexagon must have positive side length.
- 1Side length must be positive — returns error.
Introduction
A regular hexagon has six equal sides and six equal interior angles. It is one of the most useful polygons in mathematics because it divides naturally into six equilateral triangles. This calculator returns area, perimeter, apothem, circumradius, and both common diagonals from a single side-length input.
Hexagon Area Formula Explained
A regular hexagon can be split into six equilateral triangles, each with side length a.
Area of one equilateral triangle = (√3/4)a²
Multiply by 6 to get total area = (3√3/2)a²
Numeric coefficient ≈ 2.598076
Area scales with a²
The formula assumes a regular hexagon
Apothem and Circumradius
Regular hexagons have especially elegant radius relationships.
Circumradius equals side length exactly: R = a
Apothem = a√3/2
The apothem is also the height of one equilateral triangle section
These relationships make hexagons easy to model in CAD and tiling
The center-to-vertex geometry comes from 60° central angles
Hexagon Diagonals
A regular hexagon has two especially common diagonal lengths.
Long diagonal connects opposite vertices: 2a
Short diagonal skips one vertex: a√3
Long diagonal is twice the circumradius
Short diagonal matches the diameter of the inscribed equilateral-triangle circle geometry
These lengths are useful in layout and fabrication
How to Use This Calculator
Enter one positive side length. All outputs are returned instantly in the same unit system.
Use centimeters for small parts, meters for architecture, or inches for fabrication
Perimeter stays in the original unit
Area is returned in square units
Diagonal outputs are convenient for bracing and cross-span measurements
Apothem helps when fitting the hexagon inside circular or flat-sided boundaries
Worked Examples
Known values are easy to verify because the formulas are compact.
a = 1 → area ≈ 2.598076, apothem ≈ 0.866025
a = 2 → area ≈ 10.392305, perimeter = 12
a = 4 → area ≈ 41.569219, short diagonal ≈ 6.928203
a = 7 → circumradius = 7, long diagonal = 14
Doubling a doubles perimeter and diagonals but quadruples area
Why Hexagons Matter
Hexagons appear across science, engineering, and nature because they combine symmetry with efficient packing.
Honeycombs and cellular structures
Floor tiling and modular design
Mechanical parts such as nuts and bolt heads
Board games and map grids
Chemistry diagrams such as benzene rings
Comparing the Hexagon to Other Polygons
The regular hexagon sits at a useful midpoint between simple low-sided polygons and near-circular many-sided polygons.
It encloses more area than a pentagon with the same side length
It encloses less area than a heptagon or octagon with the same side length
Unlike the heptagon, the regular hexagon tessellates the plane perfectly
Its 120° interior angle makes it ideal for efficient tiling
Its construction is straightforward from a circle because R = a
FAQs
What is the area formula for a regular hexagon?
The area is A = (3√3/2)a², where a is the side length.
What is the perimeter of a regular hexagon?
Perimeter is 6 times the side length: P = 6a.
Is the circumradius equal to the side length?
Yes. For a regular hexagon, the circumradius equals the side length exactly.
What is the apothem of a regular hexagon?
The apothem is a√3/2, which is about 0.866025 times the side length.
What are the two main diagonals of a hexagon?
The long diagonal is 2a and the short diagonal is a√3.
Can a regular hexagon tile the plane?
Yes. Regular hexagons tessellate perfectly without gaps, which is one reason they appear in honeycomb structures and grid systems.
Why is the hexagon so common in geometry?
It combines symmetry, efficient area coverage, and simple formulas derived from equilateral triangles.
Do the formulas work for irregular hexagons?
No. These formulas are only for regular hexagons with equal sides and equal angles.
How does the area change if I double the side length?
Area depends on a², so doubling the side length makes the area four times larger.