Hexagon Calculator
Calculate every property of a regular hexagon — area, perimeter, apothem, circumradius, and both diagonal lengths — starting from whichever single measurement you already know, with every formula and step shown.
Regular Hexagon Calculator
What Is a Hexagon Calculator?
A hexagon is a six-sided polygon, and a regular hexagon — the shape this calculator works with — has all six sides equal in length and all six interior angles equal, each measuring exactly 120 degrees. Regular hexagons show up constantly in nature and design because of a genuinely useful geometric property: they tile a flat plane perfectly, with no gaps and no overlaps, while using less total perimeter to enclose a given area than a square grid would. That efficiency is exactly why honeycomb cells, many engineered materials, floor and wall tiling patterns, and countless mechanical and architectural designs favor the hexagon over other shapes.
A regular hexagon has several interconnected measurements, and knowing any one of them is enough to derive all the others, because they're all fixed multiples of the side length. The side length is the length of each of the six equal edges. The perimeter is simply six times the side length, since all sides are equal. The area follows from a closed-form formula involving the side length squared and √3. The apothem (also called the inradius) is the distance from the hexagon's center to the midpoint of any side — it's the radius of the largest circle that fits entirely inside the hexagon. The circumradius is the distance from the center to any vertex — and for a regular hexagon specifically, this distance is always exactly equal to the side length, a special property that doesn't hold for other regular polygons. And a regular hexagon has two distinct diagonal lengths: the short diagonal, connecting vertices that are two apart, and the long diagonal, connecting opposite vertices through the center — the long diagonal is always exactly twice the side length.
Because all of these measurements are tied together by fixed ratios, a hexagon calculator's real value is letting you start from whichever single measurement you actually have — maybe you measured a side with a ruler, maybe you know the area from a spec sheet, maybe you measured the distance across a hex bolt head (a diagonal) — and instantly derive every other property without manually working through the trigonometry each time. This kind of calculation comes up constantly in real, practical contexts: mechanical engineers and machinists sizing hex bolts, nuts, and fittings (where the "across-flats" and "across-corners" measurements correspond to the apothem-based and diagonal-based dimensions); architects and designers working with hexagonal tiling, paving, or honeycomb structural patterns; students and teachers working through regular polygon geometry; and hobbyists and crafters working with hexagonal grids, quilting patterns, or tabletop game boards.
This calculator covers all five practical starting points. Choose Side Length when you know an edge measurement directly. Choose Area when you know the enclosed surface area and need to work backward to the dimensions. Choose Perimeter when you know the total boundary length. Choose Apothem when you know the inscribed-circle radius — common in bolt and fitting specifications given as an "across-flats" measurement. And choose Diagonal when you know either the short or long diagonal — the long diagonal corresponds to an "across-corners" bolt measurement, and it's also the easiest single measurement to take with calipers across a hexagonal object.
Every mode calculates live as you type and returns the complete set of hexagon properties — side length, perimeter, area, apothem, circumradius, and both diagonals — along with the formula used, so you always see exactly how each derived value was calculated. That combination of flexibility and transparency makes this tool useful for engineers and machinists working with hexagonal hardware, architects and designers working with hexagonal patterns, students learning regular polygon geometry, and anyone who needs to go from one hexagon measurement to the complete geometric picture.
The Complete Guide to Regular Hexagons
Everything students, engineers, and designers ask about hexagon geometry and how its measurements relate.
What is a regular hexagon?
A regular hexagon is a six-sided polygon with all sides equal in length and all interior angles equal, each measuring 120 degrees. Its regularity means every one of its measurements — perimeter, area, apothem, circumradius, and both diagonals — can be derived directly from a single known value.
Side length
The side length is the length of each of the hexagon's six equal edges. It's the most fundamental measurement, since every other property scales directly from it.
Perimeter
The perimeter of a regular hexagon is simply six times the side length: Perimeter = 6 × Side Length, since all six sides are equal.
Area
The area of a regular hexagon is given by Area = (3√3/2) × Side Length², a formula derived by splitting the hexagon into six equilateral triangles and summing their areas. This closed-form formula makes area calculation exact rather than requiring a numerical approximation.
The apothem (inradius)
The apothem is the perpendicular distance from the hexagon's center to the midpoint of any side — equivalently, the radius of the largest circle that fits entirely inside the hexagon. For a regular hexagon, Apothem = (√3/2) × Side Length.
The circumradius
The circumradius is the distance from the center to any vertex. For a regular hexagon specifically, the circumradius always equals the side length exactly — a unique property among regular polygons, arising because a regular hexagon can be divided into exactly six equilateral triangles meeting at the center.
Short and long diagonals
A regular hexagon has two distinct diagonal lengths. The short diagonal connects two vertices separated by one other vertex, with length Side Length × √3. The long diagonal connects two directly opposite vertices, passing through the center, with length 2 × Side Length — exactly twice the side length, and also equal to twice the circumradius.
Why hexagons tile efficiently
Regular hexagons are one of only three regular polygons (along with triangles and squares) that can tile a flat plane with no gaps or overlaps. Among these tiling shapes, the hexagon encloses the most area for a given total perimeter, which is the geometric reason honeycomb structures — built by natural selection to maximize storage capacity per unit of wax — settled on hexagonal cells.
Interior angles
Each interior angle of a regular hexagon measures exactly 120 degrees, and the sum of all six interior angles is 720 degrees, consistent with the general polygon interior-angle-sum formula (n − 2) × 180° for n = 6.
Hexagons in mechanical engineering
Hex bolts, nuts, and fittings use hexagonal geometry specifically because a six-sided head or nut gives a wrench more contact surface and a smaller turning angle between wrench repositions than a square head would, while still being compact. Bolt specifications commonly reference either the "across-flats" measurement (equivalent to twice the apothem) or the "across-corners" measurement (equivalent to the long diagonal).
Hexagons in architecture and design
Hexagonal tiling patterns appear in flooring, paving, and wall treatments both for their visual appeal and their efficient, gapless coverage. Architects and designers calculating material needs for hexagonal tile layouts rely on the same area and perimeter formulas covered here.
Hexagons in nature
Beyond honeycombs, hexagonal or near-hexagonal patterns appear in basalt column formations, certain crystal structures, and the packing patterns of bubbles and cells — all cases where a system settles into the most space-and-material-efficient regular tiling shape available.
Converting between hexagon measurements
Because every hexagon property is a fixed multiple of the side length, converting between any two measurements is always a two-step process: solve for the side length from whichever measurement is known, then compute the target measurement from that side length using its own formula.
Common hexagon calculation mistakes
The most frequent error is confusing the short and long diagonal, which differ by a factor of √3 (roughly 1.73), not a factor of 2. Confusing the apothem with the circumradius — these are different values (apothem is always shorter) — is another common mix-up. Using an approximate value of √3 rather than its full precision compounds small errors across multi-step conversions. Applying a square or general-polygon area formula instead of the hexagon-specific one, mismeasuring across-flats versus across-corners on a physical hex object, and rounding too early in a multi-step conversion round out the most common mistakes.
Best practices for hexagon calculations
Always identify clearly which measurement you're starting from before converting, since side length, apothem, circumradius, and both diagonals are all different values that are easy to mix up. When measuring a physical hexagonal object like a bolt head, be clear about whether you're measuring across flats (apothem-based) or across corners (diagonal-based), since these give meaningfully different numbers. And keep full decimal precision through intermediate steps, rounding only the final displayed result.
How the Calculator Works
Four steps from one known measurement to the complete hexagon property set.
Choose Your Known Measurement
Select side length, area, perimeter, apothem, or diagonal — whichever you already have.
Enter the Value
Type in your measurement, and for diagonal mode, specify whether it's the short or long diagonal.
Automatic Geometric Calculation
The calculator solves for the side length first, then derives every other hexagon property from it.
Review the Full Property Set
See area, perimeter, apothem, circumradius, and both diagonal lengths, with the formula used.
Hexagon Formulas
The core equations behind every property on this page (a = side length).
Area
Perimeter
Apothem (Inradius)
Circumradius
Short Diagonal
Long Diagonal
Step-by-Step Examples
Eight worked calculations covering every mode on this page.
Example 1 — From Side Length
Example 2 — Apothem from Side Length
Example 3 — From Area
Example 4 — From Perimeter
Example 5 — From Apothem (Hex Bolt Across-Flats)
Example 6 — From Long Diagonal (Across-Corners)
Example 7 — From Short Diagonal
Example 8 — Full Property Set
Quick Reference Tables
Common hexagon dimensions and how each property relates to the side length.
Hexagon Properties by Side Length
| Side Length | Perimeter | Area |
|---|---|---|
| 1 | 6 | 2.60 |
| 5 | 30 | 64.95 |
| 10 | 60 | 259.81 |
| 20 | 120 | 1,039.23 |
Relationship Between Hexagon Measurements
| Property | In Terms of Side (a) |
|---|---|
| Perimeter | 6a |
| Apothem | 0.866a (√3÷2) |
| Circumradius | 1.000a |
| Short Diagonal | 1.732a (√3) |
| Long Diagonal | 2.000a |
Regular Polygon Comparison
| Shape | Interior Angle |
|---|---|
| Triangle | 60° |
| Square | 90° |
| Pentagon | 108° |
| Hexagon | 120° |
| Octagon | 135° |
Benefits of Using This Calculator
Why engineers, designers, and students rely on a dedicated hexagon tool.
Instant Calculations
Every mode updates live as you type — no manual trigonometry required.
Five Starting Points
Start from side, area, perimeter, apothem, or either diagonal — whichever you know.
Complete Property Set
Get every hexagon measurement at once, not just the one you asked for.
Engineering-Ready
Handles across-flats and across-corners conversions used for hex bolts and fittings.
Educational Transparency
Every result shows the formula used, not just the final number.
Mobile-Friendly Interface
Every field and result card is fully responsive across phones, tablets, and desktops.
Practical Applications
Real situations where regular hexagon geometry comes up.
- Mechanical engineering — size hex bolts, nuts, and fittings correctly.
- Machining — convert across-flats and across-corners measurements for hex stock.
- Architecture and design — plan hexagonal tile, paving, or panel layouts.
- Structural engineering — work with honeycomb-pattern structural panels.
- Mathematics education — teach and practice regular polygon geometry.
- Game and board design — build hexagonal grid-based game boards and maps.
- Crafting and quilting — calculate hexagon piece dimensions for patterns.
- Jewelry and product design — design hexagonal faceted or paneled pieces.
- Landscaping — plan hexagonal paver or garden bed layouts.
- Packaging design — design hexagonal containers or packing layouts.
- 3D printing and CAD — specify precise hexagonal model dimensions.
- General geometry problems — quickly solve any regular hexagon measurement question.
Common Mistakes
Avoid these errors when calculating hexagon measurements.
Confusing short and long diagonal
They differ by a factor of √3 (about 1.73), not a factor of 2 — mixing them up gives a meaningfully wrong side length.
Confusing apothem and circumradius
The apothem is always shorter than the circumradius — treating them as equal produces incorrect derived values.
Using an approximated √3
Rounding √3 too early (e.g. to 1.7) introduces small but compounding errors across multi-step conversions.
Applying the wrong area formula
A general or square-based area formula doesn't apply to a regular hexagon — use the hexagon-specific (3√3÷2)a² formula.
Mismeasuring across-flats vs. across-corners
These give different numbers on a physical hex object — across-flats relates to the apothem, across-corners to the diagonal.
Rounding too early
Rounding an intermediate side-length value before deriving other properties compounds small errors.
Assuming an irregular hexagon behaves the same way
These formulas apply only to regular hexagons — irregular hexagons need shape-specific calculations.
Mixing units mid-calculation
Entering some measurements in inches and others in millimeters without converting invalidates the whole result.
Frequently Asked Questions
Fifteen common questions about regular hexagon geometry, answered directly.
What is a regular hexagon?
A six-sided polygon with all sides equal in length and all interior angles equal to 120 degrees.
How do I calculate the area of a hexagon?
Use Area = (3√3 ÷ 2) × Side Length². For example, a side of 10 gives an area of approximately 259.81.
How do I calculate the perimeter of a hexagon?
Multiply the side length by 6, since all six sides are equal: Perimeter = 6 × Side Length.
What is the apothem of a hexagon?
The distance from the center to the midpoint of any side, calculated as Apothem = (√3 ÷ 2) × Side Length.
What is the circumradius of a hexagon?
The distance from the center to any vertex — for a regular hexagon, this always equals the side length exactly.
What is the difference between the short and long diagonal?
The short diagonal connects vertices two apart (length = side × √3); the long diagonal connects opposite vertices through the center (length = 2 × side).
Why does a hexagon's circumradius equal its side length?
Because a regular hexagon can be divided into six equilateral triangles meeting at the center, each with sides equal to the hexagon's side length.
Can I calculate hexagon properties from the area alone?
Yes — the Area mode above solves for the side length first, then derives every other property from it.
Is this calculator useful for hex bolts?
Yes — the Apothem mode corresponds to an across-flats measurement, and the Diagonal mode (long) corresponds to an across-corners measurement.
Why do hexagons tile so efficiently?
Regular hexagons are one of only three regular polygons that tile a plane with no gaps, and among those, they enclose the most area per unit of perimeter.
Is this calculator accurate?
Yes — it applies exact regular-hexagon geometric formulas using full-precision √3, with the formula shown alongside every result.
Is it free to use?
Yes, all five calculator modes are completely free with no sign-up required.
Does it work on mobile?
Yes — the layout and all input fields are fully responsive across phones, tablets, and desktops.
Does this work for irregular hexagons?
No — these formulas apply specifically to regular hexagons, where all sides and angles are equal.
What are the interior angles of a hexagon?
Each interior angle of a regular hexagon measures 120 degrees, summing to 720 degrees in total.
Calculate Every Hexagon Property Instantly
Find the area, perimeter, apothem, circumradius, and diagonals of any regular hexagon using the calculator above.