Regular hexagon area:
formula and worked example

Why this shape keeps showing up

Regular hexagons tile a surface with no gaps, which is why you find them in honeycomb, paving slabs, bathroom floor tiles, bolt heads and nuts, and some gazebo floor plans. Whenever a regular hexagon is the unit of a pattern, its area is the building block for estimating how many fit into a region or how much material each consumes.

Its geometry is unusually tidy. The six corners lie on a circle whose radius equals the side length, and the shape splits into six identical equilateral triangles meeting at the centre.

Hexagons are also the shapes that geometers use to explain efficiency. For a given edge length, a regular hexagon covers about 2.6 times the side squared, compared with 1 for a square and about 0.43 for an equilateral triangle.

The formula and its single input

Area equals three times the square root of three, times the side squared, divided by two. Numerically, 3 x root 3 / 2 is about 2.598, so the area is approximately 2.598 x s squared. The side s is the length of one edge, corner to adjacent corner.

Where does the constant come from? Each of the six triangles has area root 3 / 4 x s squared. Six of them gives 6 x root 3 / 4 = 3 x root 3 / 2 times s squared.

A hexagon with side 4

Consider a tile with each side measuring 4 units.

1. 1Square the side: 4 x 4 = 16.
2. 2Multiply by 3 x root 3: 16 x 3 x 1.73205 = 83.138.
3. 3Divide by two: 83.138 / 2 = 41.569.
4. 4Add the unit: 41.569 square units.

A second route confirms it. One equilateral triangle of side 4 has area root 3 / 4 x 16 = 6.928; times six is 41.569. A third check uses perimeter and apothem: the apothem is s x root 3 / 2 = 3.464 and half of perimeter (24) times apothem is 12 x 3.464 = 41.569.

Using the area

Once you have the area per hexagon, quantity planning is simple: divide the region by this area and allow wastage for edges where whole hexagons do not fit. Because hexagons interlock, a regular hexagonal tiling has no gaps in the interior, but the perimeter of any room produces cut tiles.

It also helps to compare with other shapes of equal perimeter. A regular hexagon of side 4 has perimeter 24 and area 41.57, while a square with perimeter 24 has side 6 and area 36, and a circle with the same perimeter has area about 45.8. Among simple regular shapes, the hexagon sits close to the circle, which is one reason nature favours it for packing.

For reference, the inscribed circle of a regular hexagon has radius s x root 3 / 2, so a hexagonal slab of side 4 fits a circle of radius 3.46 inside it and sits inside a circle of radius 4. This helps when a round object, such as a pot or post, has to fit inside a hexagonal frame.

Growing the side by 10%

At a side of 4.4 the area becomes 50.299, up 21% from 41.569. Because the side is squared, any uniform scale-up follows the square law: double the side and you get four times the area, and a hexagon with side 8 has 166.3.

For product sizing the lesson is that tiles that look only slightly bigger consume a lot more material per piece. A 10% step in side length costs about a fifth more surface area, which affects both the price per tile and the number you need.

Where hexagon calculations go wrong

• Measuring across flats (between two parallel sides) and treating it as the side. Across-flats is s x root 3, about 1.732 s, so for s = 4 it is 6.93, not 4.
• Taking the corner-to-corner distance as the side. That distance is 2s, twice the side.
• Dropping the halving and returning 3 x root 3 x s squared, which doubles the area.
• Applying the formula to a hexagon that is not regular, for which angles or sides differ and a split into triangles is needed.

Scenario: hexagonal patio slabs

You plan a patio of regular hexagonal slabs of side 20 cm. Each slab has area 2.598 x 400 = 1,039 square centimetres, about 0.104 square metres. A 10 square metre patio then needs 10 / 0.1039 = 96.2, so 97 slabs, and with around 10% extra for cuts at the borders, about 107 in total. The across-flats width of each slab is 34.6 cm, which you also need for sizing the edge restraints and for setting out rows.

If the slabs are laid in a pattern with a joint of a few millimetres, remember that the effective side is the slab side plus half the joint, so large patios can show a measurable gap between nominal and actual coverage.