Hexagon Calculator

How to use the hexagon calculator

The calculator is for a regular hexagon, a six-sided figure whose sides are all equal and whose angles are all 120°. Because of that symmetry, a single length fixes everything else.

1. In Solve a hexagon by, choose the quantity you know: side length, long diagonal, short diagonal, apothem, radius, area or perimeter.
2. Enter its value in the next row and pick the unit you measured it in.
3. Choose Output units. All results will be shown in this unit, and the area in the matching square unit.
4. Press Solve hexagon. You get seven lines: Area, Perimeter, Side (a), Diagonal (d), Short diagonal (s), Radius (R) and Apothem (r).

Lengths are converted from the unit you entered to the output unit before anything is computed. Results are rounded to four decimal places; whole numbers appear without decimals and values with one or two decimals are padded to two, so 0.5 prints as 0.50. When you solve by area, the unit beside the number is read as a square unit (m means square meters), and it is converted to the square of the output unit, so 100 m² shown in centimeters becomes 1,000,000 cm².

Hexagon formulas

Every formula below is written with the side length a. Together they let you move from any one measurement to all the others.

Circumradius and inradius of a hexagon

A regular hexagon has two circles attached to it. The circumscribed circle passes through all six corners, and its radius R, the circumradius, is exactly the side length. That is a convenient property: a pair of compasses set to the side length will step round the circle in six equal jumps and mark the six corners.

The inscribed circle touches the middle of every side. Its radius is the apothem r, the shortest distance from the centre to a side. It is about 86.6% of the circumradius. The ring between the two circles is thin: for a hexagon with side 10, R = 10 and r ≈ 8.66.

Example hexagon geometry problems

Example 1: from the side. A hexagonal tile has sides of 8 cm. Area = 2.598076 × 64 ≈ 166.2769 cm², perimeter = 6 × 8 = 48 cm, long diagonal = 16 cm, short diagonal = 8√3 ≈ 13.8564 cm, and apothem = 4√3 ≈ 6.9282 cm.

Example 2: from the apothem. A hexagonal gazebo floor has an apothem of 10 ft. The side is a = 2r ÷ √3 = 20 ÷ 1.732051 ≈ 11.5470 ft, the perimeter is 6 × 11.5470 ≈ 69.2820 ft, and the area is 2√3 × 10² ≈ 346.4102 ft².

Example 3: from the area. A hexagonal flower bed must cover 150 m². Then a = √(2 × 150 ÷ (3√3)) = √57.7350 ≈ 7.5984 m, and the edging needed is the perimeter, 6 × 7.5984 ≈ 45.5901 m.

Example 4: a hex nut. A spanner fits across the flats of a nut that measures 24 mm. That distance is the short diagonal, so the side is 24 ÷ √3 ≈ 13.8564 mm and the distance across the corners, the long diagonal, is 2 × 13.8564 ≈ 27.7128 mm. The nut face has an area of about 498.8306 mm² before the hole is subtracted.

Practical application

• Fasteners and tools: nuts, bolt heads and socket keys use hexagons because a spanner can grip them from six angles in a small swing.
• Tiling and paving: regular hexagons tile a flat surface with no gaps, and compared with squares or triangles they enclose a given area with less edge length.
• Nature: honeycomb cells are hexagonal, and the carbon atoms in graphene sit on a hexagonal lattice.
• Design and games: board games, maps divided into hex grids, planters, gazebos and architectural floor plans.

For the circle through the corners or inside the sides, see the area of a circle calculator. The six triangles in the hexagon can be checked with the area of a triangle calculator, and the nearby shapes have their own tools: rectangle, square, trapezoid and parallelogram. Two trapezoids of equal size also make up a regular hexagon when cut along a long diagonal.

Common mistakes

• Confusing the two diagonals. The long diagonal is twice the side; the short one is about 1.732 times the side.
• Mixing up radius and apothem. The circumradius reaches a corner; the apothem reaches the middle of a side and is shorter.
• Using an irregular hexagon. All formulas here assume equal sides and equal angles. A stretched or uneven hexagon needs to be split into triangles or trapezoids.
• Reading the wrong unit for the area. Output units apply to every line, and the area is in the square of that unit.