Hexagon Calculator (Regular Hexagon)
This hexagon calculator works out every measure of a regular hexagon from any single one: side, perimeter, area, inradius, width across flats or the long diagonal. Example: a side of 5 gives a perimeter of 30, an area of 64.9519 and a width across flats of 8.6603.
Example: Known value: side a · Value 5 → Result: Side a = 5 · area A = 64.9519 · perimeter U = 30. Source: Weisstein, Eric W.: Hexagon. MathWorld – A Wolfram Web Resource (inradius, circumradius, area of the regular hexagon). Updated: .
How it is calculated
A regular hexagon has six equal sides and six interior angles of 120°. It consists of six equilateral triangles that meet in the centre – which is why its circumradius is exactly as long as its side. One known measure is therefore enough to calculate all the others.
Formulas
- Perimeter: U = 6 · a
- Area: A = (3 · √3 ÷ 2) · a² ≈ 2.598076 · a²
- Long diagonal (corner to corner): d = 2 · a
- Short diagonal = width across flats: d₂ = √3 · a ≈ 1.732051 · a
- Inradius (apothem): r = (√3 ÷ 2) · a; circumradius: R = a
The area formula follows directly from the six triangles: each has the area (√3 ÷ 4) · a², and six of them give (3 · √3 ÷ 2) · a². Equally, A = U · r ÷ 2.
Worked example
Side a = 5 cm: perimeter U = 6 × 5 = 30 cm. Area A = 2.598076 × 25 = 64.9519 cm². Long diagonal d = 10 cm, width across flats d₂ = 1.732051 × 5 = 8.6603 cm, inradius r = 4.3301 cm.
Working backwards
If the area is known, solve for the side: a = √(A ÷ 2.598076). An area of 100 m² gives a = √38.49 = 6.2040 m. For a hexagonal nut or bolt head, the spanner size is the width across flats: with 17 mm across flats the side is 17 ÷ 1.732051 = 9.8150 mm and the corner-to-corner dimension is 19.6299 mm – the clearance the head needs to turn.
Where hexagons turn up
Hexagons tile a surface without gaps, so they appear in honeycombs, paving stones, floor tiles and board-game maps. For a tiled area, calculate the area of one tile with the formula above and divide the total area by it; allow extra for cuts at the edges.
Frequently asked questions
How do I calculate the area of a hexagon?
For a regular hexagon, multiply the square of the side by 2.598076 (that is 3 · √3 ÷ 2). With a side of 5 the area is 2.598076 × 25 = 64.9519.
How do I find the side from the area?
Divide the area by 2.598076 and take the square root. An area of 100 gives a side of 6.2040.
What is the difference between the long and the short diagonal?
The long diagonal runs from corner to opposite corner through the centre and is twice the side. The short diagonal joins two corners with one corner between them; it equals the distance between two opposite sides, √3 times the side. For a side of 5 they are 10 and 8.6603.
What are the angles of a regular hexagon?
Each interior angle is 120°, because the angle sum of a hexagon is (6 − 2) × 180° = 720°. Each exterior angle and each central angle is 60°.
Does this work for irregular hexagons?
No. With unequal sides or angles, a single measure is not enough. Split the shape into triangles and add up their areas, or use coordinates.
Sources and legal basis
- Weisstein, Eric W.: Hexagon. MathWorld – A Wolfram Web Resource (inradius, circumradius, area of the regular hexagon)
- Weisstein, Eric W.: Regular Polygon. MathWorld – A Wolfram Web Resource (Innenwinkel, Fläche = Umfang × Inkreisradius ÷ 2)
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