MATHEMATICS2D GeometryMathematics Calculator
๐Ÿ”ข

Hexagon

Calculate the area, perimeter, inradius, circumradius, and height of a regular hexagon from any known dimension. Supports side length, area, perimeter, inradius, or circumradius input with step-by-...

Did our AI summary help? Let us know.

Why: Understanding hexagon helps you make better, data-driven decisions.

How: Enter Unit to calculate results.

Run the calculator when you are ready.

Start CalculatingExplore mathematical calculations
โฌก
GEOMETRY2D Shapes

Hexagon โ€” Nature's Perfect Shape

Calculate any property from just one measurement. 5 input modes: side length, area, perimeter, inradius, or circumradius.

Square โ†’Pentagon โ†’Octagon โ†’Polygon โ†’

โฌก Real-World Examples โ€” Click to Load

Calculation Settings

cm

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿ“‹ Key Takeaways

  • โ€ข A regular hexagon has 6 equal sides and 6 interior angles of 120ยฐ โ€” nature's most efficient tessellating shape
  • โ€ข The circumradius equals the side length: R = s โ€” a unique property among regular polygons
  • โ€ข The inradius (apothem) is (โˆš3/2)s โ‰ˆ 0.866 ร— side, and height = 2 ร— inradius
  • โ€ข Hexagons tessellate perfectly with no gaps โ€” honeybees use this for maximum storage with minimum wax
  • โ€ข Only one measurement (side, area, perimeter, inradius, or circumradius) is needed to calculate everything

๐Ÿ’ก Did You Know?

๐ŸHoneybees build hexagonal cells because hexagons use the least wax while maximizing storage โ€” a perfect example of nature's mathematical optimizationSource: Biology & Geometry
โš—๏ธThe benzene molecule (Cโ‚†Hโ‚†) has a perfect hexagonal carbon ring structure โ€” foundational in organic chemistrySource: Chemistry
๐Ÿ”๏ธGiant's Causeway in Northern Ireland features ~40,000 interlocking hexagonal basalt columns formed by volcanic coolingSource: Geology
๐Ÿ›‘US stop signs are octagons, but many countries use hexagonal warning signs. The hexagon appears in traffic signage worldwideSource: Transportation
๐Ÿ”ฉHexagonal bolt heads allow 6 contact points for wrenches โ€” optimal grip and stress distribution in mechanical engineeringSource: Engineering
๐ŸŒ€Saturn's north pole has a persistent hexagonal storm pattern โ€” a natural hexagon roughly 30,000 km acrossSource: Planetary Science

๐Ÿ“– Hexagon Formulas Explained

Area of a Regular Hexagon

The area is found by dividing the hexagon into 6 equilateral triangles from the center. Each triangle has area (โˆš3/4)sยฒ, so total area = 6 ร— (โˆš3/4)sยฒ = (3โˆš3/2)sยฒ.

A=332s2โ‰ˆ2.598ร—s2A = \frac{3\sqrt{3}}{2}s^2 \approx 2.598 \times s^2

where s = side length

Example: s = 10 cm โ†’ A โ‰ˆ 259.8 cmยฒ.

6 triangles

Perimeter of a Regular Hexagon

The perimeter is the sum of all six equal sides.

P=6sP = 6s

Example: s = 10 cm โ†’ P = 60 cm.

P = 6s

Inradius & Circumradius

Inradius (apothem) r = distance from center to side. Circumradius R = distance from center to vertex. For a regular hexagon: R = s (unique!), r = (โˆš3/2)s.

r=32s,R=sr = \frac{\sqrt{3}}{2}s, \quad R = s

Height between parallel sides: h = 2r.

R=sr

๐ŸŽฏ Expert Tips

๐Ÿ’ก Hexagonal Tiling

Hex tiles fit together with no gaps. When tiling a floor, add 10โ€“15% for cuts. For a 10 mยฒ room with 500 cmยฒ tiles: 10 รท 0.05 = 200 tiles + 15% โ‰ˆ 230 tiles.

๐Ÿ’ก Circumradius = Side

In a regular hexagon, R = s. If you measure corner-to-corner through the center (diameter = 2R), the side equals half that distance. Quick check for hex bolts!

๐Ÿ’ก Area Scaling

Area scales by sยฒ. Double the side โ†’ 4ร— the area. A hexagon with side 2s has area 4 times that of side s.

๐Ÿ’ก Inradius for Layout

The inradius is the distance from center to the midpoint of any side. Useful for hexagonal gazebos, tables, or garden beds โ€” measure from center to edge to get r, then derive all dimensions.

โš–๏ธ Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorBasic CalculatorManual
5 input modes (side/area/P/r/R)โœ…โŒโŒ
Step-by-step solutionsโœ…โŒโœ…
Interactive chartsโœ…โŒโŒ
Copy & share resultsโœ…โŒโŒ
All properties at onceโœ…โš ๏ธ Limitedโš ๏ธ Tedious
Unit support (mm to mi)โœ…โš ๏ธ LimitedโŒ
Real-world examplesโœ…โŒโŒ
Mobile-friendlyโœ…โš ๏ธ VariesโŒ

๐Ÿ“Š Hexagon Quick Facts

6
Equal Sides
120ยฐ
Interior Angle
6
Symmetry Lines
60ยฐ
Exterior Angle

โ“ Frequently Asked Questions

How do I find the area of a hexagon?

For a regular hexagon: A = (3โˆš3/2)sยฒ โ‰ˆ 2.598sยฒ. If you know the side length s, plug it in. If you know area and need side: s = โˆš(A / (3โˆš3/2)).

What is the difference between a regular and irregular hexagon?

A regular hexagon has all 6 sides and all 6 angles equal (each 120ยฐ). An irregular hexagon has 6 sides but they can differ in length and angle. This calculator works only for regular hexagons.

Why does circumradius equal side length in a hexagon?

In a regular hexagon, the center-to-vertex distance (circumradius) equals the side length because the hexagon can be inscribed in a circle where each side subtends 60ยฐ โ€” the sides and radii form equilateral triangles.

What is the apothem of a hexagon?

The apothem (inradius) is the perpendicular distance from the center to any side. For a regular hexagon: r = (โˆš3/2)s โ‰ˆ 0.866s. It's the radius of the inscribed circle.

How does hexagon area compare to a square with the same side?

A regular hexagon has area (3โˆš3/2)sยฒ โ‰ˆ 2.598sยฒ, while a square has sยฒ. So the hexagon has about 2.6ร— the area of a square with the same side length.

Why do honeybees use hexagonal cells?

Hexagons tessellate with no gaps and use the least wax for a given storage volume. It's the most efficient shape for honeycomb โ€” nature's optimization.

How do I find the side from the inradius?

Since r = (โˆš3/2)s, we get s = 2r/โˆš3. For example, if r = 3 m, then s = 2ร—3/โˆš3 โ‰ˆ 3.46 m.

What is the height of a regular hexagon?

The height (distance between parallel sides) equals 2 ร— inradius = 2r. So h = โˆš3 ร— s โ‰ˆ 1.732s. Useful when the hexagon is oriented with a flat side on top.

โš ๏ธ Disclaimer: This calculator provides mathematically precise results based on standard Euclidean geometry formulas for a perfect regular hexagon. Real-world measurements may vary due to manufacturing tolerances, measurement precision, and material properties. Always verify measurements for construction and engineering projects.

๐Ÿ‘ˆ START HERE
โฌ…๏ธJump in and explore the concept!
AI

Related Calculators

Ellipse Calculator

Calculate area, perimeter, eccentricity, and other properties of ellipses with our free online calculator.

2D Geometry

Golden Rectangle Calculator

Calculate dimensions, area, and other properties of golden rectangles based on the golden ratio (ฯ† โ‰ˆ 1.618).

2D Geometry

Octagon Calculator

Octagon Calculator - Calculate area, perimeter, and other properties of an octagon with step-by-step solutions

Mathematics

Quadrilateral Calculator

Calculate area, perimeter and other properties of quadrilaterals including squares, rectangles, rhombuses, parallelograms, trapezoids, kites, and general...

Mathematics

Cube Calculator

Cube Calculator - Calculate volume, surface area, and other properties of a cube with step-by-step solutions

Mathematics

Cylinder Calculator

Cylinder Calculator - Calculate volume, surface area, and other properties of a right cylinder with step-by-step solutions

Mathematics