Hexagon Calculator — Properties of a Regular Hexagon
Welcome to the hexagon calculator, a handy tool when dealing with any regular hexagon. The hexagon shape is one of the most popular shapes in nature, from honeycomb patterns to hexagon tiles for mirrors — its uses are almost endless. Here we explain not only why the 6-sided polygon is so popular but also how to use all the geometric properties of a regular hexagon.
How many sides does a hexagon have? Exploring the 6-sided shape
It should be no surprise that the hexagon (also known as the "6-sided polygon") has precisely six sides. This fact is true for all hexagons since it is their defining feature. The length of the sides can vary even within the same hexagon, except when it comes to the regular hexagon, in which all sides must have equal length.
The regular hexagon is the most common among 6-sided polygons and the one most often found in nature. All internal angles of a regular hexagon equal 120°, and all sides are equal.
Hexagon definition — what is a regular hexagon?
As we've already mentioned, the regular hexagon must have all sides of equal length and all its internal angles must be equal. The sum of all interior angles in any hexagon is 720°. Since all 6 angles are equal in a regular hexagon, each angle = 720° ÷ 6 = 120°.
The perimeter of a regular hexagon is simply:
P = 6 × a
where a is the side length.
Hexagon area formula — how to find the area of a hexagon
A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to a. The area of one equilateral triangle is:
A₀ = (√3 / 4) × a²
Multiplying by 6 gives the hexagon area formula:
A = 6 × A₀ = (3√3 / 2) × a²
This is equivalent to the general polygon area formula: A = apothem × perimeter / 2, where the apothem (inradius) equals r = (√3 / 2) × a.
Diagonals of a hexagon
A regular hexagon has two types of diagonals:
- Long diagonal (d) — connects opposite vertices: d = 2a
- Short diagonal (ds) — connects vertices two steps apart: ds = √3 × a
A regular hexagon has 9 diagonals in total: 3 long and 6 short.
Circumradius and inradius
Two important radii characterize a regular hexagon:
- Circumradius (R) — radius of the circumscribed circle (passes through all vertices): R = a. The circumscribed circle has the same radius as the side length — a unique property of the regular hexagon!
- Inradius (r) — radius of the inscribed circle (tangent to all sides): r = (√3 / 2) × a ≈ 0.866 × a
How to use this hexagon calculator
Simply enter the side length and select your preferred unit (metric: mm, cm, m, km — or imperial: in, ft, yd, mi). The calculator instantly computes:
- Area
- Perimeter
- Circumradius (R) — circumscribed circle radius
- Inradius (r) — inscribed circle radius (apothem)
- Long diagonal
- Short diagonal
- Interior angle (always 120°)
Hexagon tiles and real-world uses of the 6-sided polygon
The regular hexagon appears everywhere in the real world:
- Honeycomb pattern — bees build their cells in a hexagonal grid because it is the most efficient way to tile a plane with the minimum total perimeter (and thus minimum wax) for a given area.
- Floor and wall tiles — hexagonal tiles cover surfaces without gaps and create visually appealing patterns.
- Snowflakes — water molecules bond in a hexagonal lattice, giving snowflakes their 6-fold symmetry.
- Nuts and bolts — hex bolts and nuts use the hexagonal shape for efficient wrench grip.
- Carbon structures — graphene consists of carbon atoms arranged in a hexagonal lattice, making it the strongest known material.
- Board games — strategy games like Catan use hexagonal tiles for even neighbor relationships in all directions.
Honeycomb pattern — why the 6-sided shape is so prevalent in nature
The regular hexagon is the shape that tiles a flat surface with equal regular polygons while minimizing total perimeter for a given area. This is known as the honeycomb conjecture, proven mathematically in 1999 by Thomas Hales. This is why bees — through millions of years of evolution — arrived at the hexagonal cell shape for their honeycombs: it uses the least wax while providing the most storage space.
FAQs
Q: What is the area of a regular hexagon with side 5 cm?
A: A = (3√3 / 2) × 5² = (3 × 1.7321 / 2) × 25 ≈ 64.95 cm²
Q: What is the perimeter of a regular hexagon with side 8 inches?
A: P = 6 × 8 = 48 inches
Q: Are all hexagons regular?
A: No. A regular hexagon has all sides and all angles equal. Irregular hexagons can have sides and angles of different lengths. This calculator works for regular hexagons only.
Q: Is the circumradius of a hexagon equal to its side?
A: Yes! For a regular hexagon, the circumscribed circle radius equals the side length: R = a.