Hexagon Calculator
Calculate area, perimeter, side length, and apothem for a regular hexagon.
Calculation Results
Enter a value to compute all properties of a regular hexagon.
Side (a): units
Perimeter (P): units
Apothem (ap): units
Area (A): square units
Long diagonal (d): units
Short diagonal (s): units
Circumcircle radius (R): units
Regular Hexagon Diagram
The Regular Hexagon
A regular hexagon is a 6-sided polygon with all sides and angles equal. It can be partitioned into 6 equilateral triangles.
- Perimeter: P = 6a
- Apothem: ap = (√3 / 2) · a
- Area: A = (1/2) · P · ap = (3√3 / 2) · a²
- Interior Angle: 120° (each)
- Long Diagonal (d): 2a
- Short Diagonal (s): a × √3
- Circumcircle Radius (R): a
Hexagon Formulas
- Perimeter: P = 6a
- Apothem: ap = (√3 / 2) · a
- Area: A = (3√3 / 2) · a² = (1/2) · P · ap
- Long Diagonal: d = 2a
- Short Diagonal: s = a × √3
- Circumcircle Radius: R = a
- Interior Angle: 120° (each)
- Exterior Angle: 60° (each)
- Side Length from Perimeter: a = P / 6
- Side Length from Area: a = √(2A / (3√3))
- Apothem from Area: ap = (2A) / P
Useful inverses:
a = P/6,
a = √(2A / (3√3)).
Side Length (a)
Apothem (ap)
Perimeter (P = 6a)
Area (A = ½ × P × ap)
How to Use the Hexagon Calculator
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Select What You Know
Choose one of: Side, Perimeter, or Area.
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Enter the Value
Provide a positive number and click Calculate.
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Read Results
The tool returns a, P, ap, and A. Use Reset to clear.
Example Calculation
- Given: a = 8 cm
- Perimeter: P = 6a = 48 cm
- Apothem: ap = (√3/2)·8 ≈ 6.9282 cm
- Area: A = (3√3/2)·8² ≈ 166.2769 cm²
Formula Derivation (Regular Hexagon)
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Perimeter
Sum of six equal sides: P = 6a.
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Apothem
A regular hexagon decomposes into 6 equilateral triangles with side a. The apothem is the height of one equilateral triangle: ap = (√3/2)·a.
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Area
Area = sum of 6 equilateral triangle areas. Each equilateral triangle has area (√3/4)·a², so A = 6·(√3/4)·a² = (3√3/2)·a². Alternatively, polygon area formula gives A = (1/2)·P·ap.
Special Cases
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Regular Hexagon
P = 6a, ap = (√3/2)·a, A = (3√3/2)·a².
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Irregular Hexagon
Perimeter = sum of sides. Area typically requires coordinates (shoelace formula) or decomposing into triangles/quadrilaterals. This calculator assumes a regular hexagon.
Hexagon Formulas (Comparison)
| Type | Perimeter (P) | Apothem (ap) | Area (A) |
|---|---|---|---|
| Regular Hexagon | 6a | (√3/2)·a | (3√3/2)·a² = (1/2)·P·ap |
| Irregular Hexagon | a₁ + … + a₆ | — | Decompose or use shoelace |
Frequently Asked Questions
A regular hexagon is a six-sided polygon where all sides and all interior angles are equal (each interior angle is 120°).
Multiply the side length (a) by 6: P = 6a.
The apothem is the distance from the center to the middle of a side. For regular hexagons: ap = (√3 / 2) × a.
You can use either: A = (3√3 / 2) × a² or A = (1/2) × Perimeter × Apothem.
A long diagonal connects two opposite vertices (spans across the shape): d = 2a. A short diagonal skips one vertex: s = a × √3.
From perimeter: a = P / 6
From area: a = √(2A / (3√3))
The outputs use the same units as your input. If you enter side length in cm, the area will be in cm², etc.