Hexagon Calculator
How a Regular Hexagon Works
A regular hexagon is a six-sided polygon with all sides of equal length and all interior angles measuring exactly 120 degrees. It is one of only three regular polygons that can tessellate a plane (tile without gaps), along with the equilateral triangle and the square. According to Wolfram MathWorld, the regular hexagon possesses D6 symmetry with 6 lines of reflective symmetry and rotational symmetry of order 6, making it one of the most symmetric two-dimensional figures. Hexagons appear throughout nature, architecture, and engineering, from honeycomb cells to bolt heads to the hexagonal cloud pattern on Saturn's north pole. Use this calculator alongside our area calculator and pentagon calculator for related polygon computations.
Hexagon Formulas Explained
All properties of a regular hexagon derive from its side length (s). The key formulas, as documented in standard geometry references such as the Khan Academy geometry curriculum, are:
- Area = (3√3 / 2) x s² ≈ 2.598 x s²
- Perimeter = 6s
- Apothem (center to midpoint of side) = s x √3 / 2 ≈ 0.866s
- Short Diagonal (skipping one vertex) = s x √3 ≈ 1.732s
- Long Diagonal (opposite vertices) = 2s
Worked example: For a hexagon with side length s = 10 units: Area = (3√3 / 2) x 100 = 259.81 square units. Perimeter = 60 units. Apothem = 8.66 units. Short diagonal = 17.32 units. Long diagonal = 20 units. The hexagon can be divided into 6 equilateral triangles, each with area 259.81 / 6 = 43.30 square units.
Key Hexagon Terms
- Regular Hexagon — a hexagon with all six sides equal and all six interior angles equal (120 degrees each). All calculations on this page assume a regular hexagon.
- Apothem — the perpendicular distance from the center to the midpoint of any side. Also the inradius (radius of the inscribed circle).
- Circumradius — the distance from the center to any vertex. For a regular hexagon, the circumradius equals the side length (R = s).
- Tessellation — a pattern of shapes that fills a plane with no gaps or overlaps. Regular hexagons tessellate because 3 hexagons meet at each vertex (3 x 120 degrees = 360 degrees).
- Interior Angle Sum — calculated as (n - 2) x 180 degrees, where n = 6. The sum is 720 degrees, divided equally into six 120-degree angles.
Regular Polygon Comparison
The table below compares key properties of regular polygons from 3 to 8 sides, showing how area efficiency increases with more sides (approaching a circle).
| Polygon | Sides | Interior Angle | Area (s=10) | Tessellates? |
|---|---|---|---|---|
| Triangle | 3 | 60 degrees | 43.3 | Yes |
| Square | 4 | 90 degrees | 100.0 | Yes |
| Pentagon | 5 | 108 degrees | 172.0 | No |
| Hexagon | 6 | 120 degrees | 259.8 | Yes |
| Heptagon | 7 | 128.6 degrees | 363.4 | No |
| Octagon | 8 | 135 degrees | 482.8 | No |
Practical Hexagon Examples
Example 1 — Honeycomb cell: A typical honeybee cell has a side length of approximately 3.2 mm. Area = 2.598 x 3.2² = 26.6 mm². The perimeter is 19.2 mm. Bees build hexagonal cells because they use the least wax (minimum perimeter) to enclose the most honey (maximum area), a principle mathematically proven by Thomas Hales in 1999 known as the honeycomb conjecture.
Example 2 — Hexagonal floor tile: A homeowner is tiling a bathroom floor with regular hexagonal tiles having 6-inch sides. Each tile covers 2.598 x 36 = 93.5 square inches = 0.65 square feet. A 50 square foot bathroom needs approximately 50 / 0.65 = 77 tiles, plus 10-15% waste for cuts and breakage.
Example 3 — Engineering bolt head: A standard M10 hex bolt has a head with an across-flats dimension of 17 mm, making the side length s = 17 / √3 = 9.81 mm. The area of the bolt head face is 2.598 x 96.3 = 250.2 mm². Engineers calculate bolt head area to determine bearing stress on clamped surfaces using our triangle area calculator for related cross-section computations.
Tips for Hexagon Calculations
- Remember: circumradius equals side length. Unlike most polygons, a regular hexagon's circumscribed circle radius is exactly equal to the side length (R = s), making center-to-vertex distances trivial to compute.
- Use the 6-triangle decomposition: Every regular hexagon can be split into 6 congruent equilateral triangles. This simplifies area calculations and makes it easy to find the apothem (triangle height).
- Convert from diagonal measurements: If you only have the long diagonal (D), the side length is s = D / 2. If you have the short diagonal (d), then s = d / √3.
- Account for orientation: Hexagons can sit flat-bottom (pointy top) or pointy-bottom. The apothem is always the distance to the flat side, regardless of orientation.
- Hexagon area from apothem: If you know the apothem (a) instead of side length, use Area = 2√3 x a², which is useful for inscribed-circle-based measurements.
Frequently Asked Questions
What is the area formula for a regular hexagon?
The area of a regular hexagon is A = (3√3 / 2) x s², where s is the side length. This equals approximately 2.598 x s². For a hexagon with side 10, the area is 259.81 square units. This formula derives from the fact that a regular hexagon consists of 6 equilateral triangles, each with area (√3 / 4) x s². Multiplying by 6 gives the full hexagon area. You can also calculate area from the apothem (a) using A = 2√3 x a², or from the perimeter (P) and apothem using A = (P x a) / 2.
How many diagonals does a hexagon have?
A regular hexagon has 9 diagonals in total, calculated using the formula n(n-3)/2 where n = 6: 6(6-3)/2 = 9. These consist of 3 long diagonals connecting opposite vertices (each with length 2s) and 6 short diagonals connecting vertices separated by one vertex (each with length s√3). The three long diagonals all pass through the center and bisect each other, dividing the hexagon into 6 equilateral triangles. This is a unique property not shared by pentagons or heptagons, making the hexagon especially easy to subdivide.
What is the interior angle of a regular hexagon?
Each interior angle of a regular hexagon measures exactly 120 degrees. The sum of all interior angles is 720 degrees, calculated using the formula (n-2) x 180 degrees where n = 6: (6-2) x 180 = 720 degrees. Since all angles are equal in a regular hexagon, each angle is 720 / 6 = 120 degrees. This 120-degree angle is what allows hexagons to tessellate: three hexagons meeting at a vertex produce 3 x 120 = 360 degrees, perfectly filling the space around a point.
Why are hexagons so common in nature?
Hexagons appear in nature because they provide the most efficient way to partition a surface into equal areas with the minimum total perimeter. This principle, proven mathematically by Thomas Hales in 1999, explains why honeybees build hexagonal cells: the shape minimizes wax usage while maximizing honey storage volume. Beyond honeycombs, hexagonal patterns appear in basalt columns (like the Giant's Causeway), turtle shells, insect compound eyes, and snowflake crystal structures. The hexagonal packing of circles is also the densest possible arrangement in two dimensions, a fact proven by Hales in 2005.
How do I find the side length from the area?
To find the side length when you know the area, rearrange the area formula: s = √(2A / 3√3) = √(A / 2.598). For example, if a hexagonal tile has an area of 150 square centimeters, the side length is √(150 / 2.598) = √57.74 = 7.60 cm. You can verify by plugging back in: 2.598 x 7.60² = 2.598 x 57.76 = 150.1 square cm. Similarly, you can derive the side from the apothem (s = 2a / √3), from the long diagonal (s = D / 2), or from the perimeter (s = P / 6).