Hexagon calculator

What is a hexagon calculator?

The hexagon calculator is an all-in-one tool for the regular hexagon, the six-sided shape with equal sides and equal angles. Enter any one of its measurements and every other quantity appears at once: the side length, the area, the perimeter, the long and short diagonals, the circumradius, and the inradius. It is useful for students working through geometry problems, makers cutting tiles or bolts, and anyone laying out a honeycomb pattern, where the regular hexagon shows up again and again because it tiles a plane with no gaps.

Properties of a regular hexagon

A regular hexagon has six equal sides and six interior angles of 120 degrees each. It can be split into six identical equilateral triangles that meet at the center, which is why its circumradius — the distance from the center to a corner — is exactly equal to the side length. The hexagon has two kinds of diagonal: three long diagonals that pass through the center and connect opposite corners, and six short diagonals that skip one corner. The long diagonal is twice the side, while the short diagonal equals the side times the square root of three.

How does the calculator work?

Type a value into any field and the calculator first recovers the side length from it, then fills in every remaining property. So you can start from the side, the area, the perimeter, either diagonal, the circumradius, or the inradius, and you will always get a complete description of the hexagon. Each length field accepts different units, and the conversions between them happen automatically.

Formulas

With side length $a$, the area of a regular hexagon is:

The perimeter is six times the side:

The long diagonal (corner to opposite corner) and the short diagonal (corner skipping one) are:

The circumradius $R$ (center to corner) equals the side, and the inradius $r$ (center to the midpoint of a side, also called the apothem) is:

where $A$ is the area, $P$ the perimeter, $D$ and $d$ the long and short diagonals, $R$ the circumradius, $r$ the inradius, and $a$ the side length.

Examples

1. A regular hexagon with a side of 10 cm:

2. Working backwards from a perimeter of 60 cm, the side is $60 / 6 = 10$ cm, which reproduces all of the values above.

Practical notes

• Because the circumradius equals the side, a regular hexagon fits perfectly inside a circle whose radius is the side length — handy when drawing one with a compass.
• The inradius is also called the apothem; it is the radius of the largest circle that fits inside the hexagon.
• For shapes with a different number of sides, the regular polygon area calculator and the regular polygon perimeter calculator generalize these formulas.

FAQs

How do I find the area of a regular hexagon?

Square the side length and multiply by $\frac{3\sqrt{3}}{2}$. For a side of 10 the area is $\frac{3\sqrt{3}}{2}\times 100 \approx 259.81$.

What is the difference between the long and short diagonals?

The long diagonal joins two opposite corners and passes through the center, so it equals $2a$. The short diagonal joins two corners separated by one vertex and equals $\sqrt{3}\,a$, which is shorter.

Why does the circumradius equal the side?

A regular hexagon splits into six equilateral triangles meeting at the center. Each triangle has the center-to-corner distance and the side as two of its equal edges, so the circumradius is exactly the side length.

What is the apothem of a hexagon?

The apothem is the inradius — the distance from the center to the middle of a side. For a regular hexagon it equals $\frac{\sqrt{3}}{2}\,a$, about 0.866 times the side.