Enter the side length of any regular hexagon and instantly find its area, perimeter, long and short diagonals, apothem, and circumradius. Full step-by-step working shown.
✓Last verified: April 2026 · Sources listed below
cm
Please enter a positive side length.
Length of one side of the regular hexagon
💡 Unique property: In a regular hexagon, the circumradius (centre to vertex) always equals the side length.
Area of Hexagon
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Sources & Methodology
✓All formulas verified against Khan Academy geometry content and standard polygon reference tables. Regular hexagon properties are classical results of Euclidean geometry.
Comprehensive reference for regular hexagon properties, diagonals, and area derivation
Methodology: Area = (3√3/2) × s². Perimeter = 6s. Apothem = (s√3)/2. Long diagonal = 2s. Short diagonal = s√3. Circumradius = s. Interior angle = 120°. π = 3.14159265358979, √3 = 1.73205080757.
⏱ Last reviewed: April 2026
Hexagon Calculator — All Formulas for 2026
A regular hexagon has six equal sides and six equal interior angles of 120° each. It is one of only three regular polygons that can tile a plane without gaps, and it holds a special place in nature and engineering due to its efficient use of area relative to perimeter. All the properties below follow from a single input: the side length s.
Key Formulas
Area = (3√3 / 2) × s² ≈ 2.598 × s²
Perimeter = 6s | Apothem = (s√3)/2 ≈ 0.866s | Long diagonal = 2s | Short diagonal = s√3 ≈ 1.732s | Circumradius = s
Reference Table: Regular Hexagon by Side Length
Side (s)
Area
Perimeter
Apothem
Long Diag
1
2.598
6
0.866
2
3
23.38
18
2.598
6
5
64.95
30
4.330
10
6
93.53
36
5.196
12
10
259.81
60
8.660
20
12
374.12
72
10.392
24
Why Are There Two Types of Diagonal?
A regular hexagon has 9 diagonals. The 3 longest pass through the centre, connecting opposite vertices at distance 2s — twice the side length. The 6 shorter diagonals skip one vertex and have length s√3. For a hexagon with s = 5: long diagonals = 10, short diagonals ≈ 8.66. These two types arise because the hexagon decomposes into 6 equilateral triangles of side s, and the diagonals correspond to different connections across these triangles.
Why Hexagons Tile So Efficiently
Among all regular polygons, the hexagon gives the greatest area for a given perimeter when used as a tiling unit. Compared to a square: a regular hexagon with the same perimeter has area ≈ 10.5% larger than the square. This is why bees use hexagonal honeycomb cells — maximum honey storage with minimum wax. It is also why hexagonal tiles are popular in floors and walls, and why carbon atoms in graphene form a hexagonal lattice.
The Circumradius Equals the Side Length
A remarkable property of regular hexagons is that the circumradius R (distance from the centre to any vertex) is exactly equal to the side length s. This is because each of the 6 equilateral triangles formed by connecting the centre to adjacent vertices has all three sides equal to s. No other regular polygon shares this property. It means that 6 regular hexagons arranged around a central point fit together exactly — visible in tiling patterns everywhere.
💡 Practical use: Hexagonal bolt heads and nuts are sized using these formulas. A bolt head with “across flats” dimension d has apothem = d/2, so side s = d/√3. Engineers calculate material volume, torque, and tool clearance using the full set of hexagon properties.
Frequently Asked Questions
Area = (3 x sqrt(3) / 2) x s squared, approximately 2.598 x s squared. For s = 6: area = 2.598 x 36 = 93.53 square units. The formula comes from dividing the hexagon into 6 equilateral triangles each with area (sqrt(3)/4) x s squared.
Perimeter = 6 x s, since a regular hexagon has 6 equal sides. For s = 7: perimeter = 42 units.
The apothem is the perpendicular distance from the centre to the midpoint of any side: apothem = (s x sqrt(3)) / 2, approximately 0.866 x s. For s = 10: apothem = 8.660.
There are two types: the long diagonal passing through the centre = 2s; the short diagonal skipping one vertex = s x sqrt(3). For s = 5: long = 10, short = 8.660.
Area = (3 x sqrt(3) / 2) x 16 = 2.598 x 16 = 41.57 square units.
Drawing lines from the centre to each of the 6 vertices divides the hexagon into 6 identical triangles. Because all sides of a regular hexagon are equal and the interior angle at each vertex is 120 degrees, each triangle has all three sides equal to s, making them equilateral. The total area is therefore 6 x (sqrt(3)/4) x s squared = (3 x sqrt(3)/2) x s squared.
Rearrange A = (3 x sqrt(3) / 2) x s squared to get s = sqrt(2A / (3 x sqrt(3))). For A = 93.53: s = sqrt(2 x 93.53 / 5.196) = sqrt(36) = 6.
The circumradius R equals the side length s exactly — R = s. This is unique to regular hexagons. For any other regular polygon, the circumradius differs from the side length.
A hexagon has 9 diagonals: 3 long diagonals (through the centre, each length 2s) and 6 short diagonals (skipping one vertex, each length s x sqrt(3)).
Each interior angle = (6 - 2) x 180 / 6 = 720 / 6 = 120 degrees. Each exterior angle = 60 degrees. The sum of all interior angles = 720 degrees.
Regular hexagons tile the plane perfectly and maximise enclosed area for a given perimeter among all regular tilings. This means bees store the most honey while using the least wax. The Honeybee Conjecture — that hexagons are the optimal shape for this — was proven mathematically by Thomas Hales in 1999.
If apothem a is known: Area = 3 x a squared x sqrt(3), or use Area = (1/2) x perimeter x apothem = 3 x s x a. Since apothem = (s x sqrt(3))/2, you can also get s = 2a / sqrt(3) then apply the area formula.
Honeycomb cells, snowflake cross-sections, hexagonal floor tiles, bolt heads and nuts, pencil cross-sections, graphene carbon lattice, basalt columns at the Giant's Causeway, and some crystal structures. Hexagons appear in engineering, biology, geology, and materials science.
No — this calculator assumes a regular hexagon where all six sides and all six angles are equal. For an irregular hexagon (unequal sides or angles) you would need to break it into triangles and sum their areas using the coordinate or Shoelace formula.