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Hexagon Calculator

Enter any one measurement of a regular hexagon (side, perimeter, area, diagonal or radius) and get all the others.

Level: Grades 7–8

  • Common Core standard 7.G.B.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
  • Common Core standard 8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Printable worksheet

Hexagon Calculator

Know the answer you need? Pick an input to work backward.

Inputs

Result

Area

41.5692 square units

A regular hexagon with side 4 has perimeter 24 and area 41.5692.

Side length (a)
4
Perimeter
24
Short diagonal
6.9282
Long diagonal
8
Inradius (apothem)
3.4641
Circumradius
4

See the step-by-step solution ↓

Step-by-step solution

  1. Perimeter: add up the six equal sides, so P = 6a

    P=6×4=24

  2. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

    A=3×32×42=41.569219

  3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

    d=3×4=6.928203

  4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

    D=2×4=8

  5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

    r=32×4=3.464102

  6. Circumradius (center to a corner) equals the side: R = a

    R=a=4

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Know one measurement of a six-sided shape, and get every other one in a click. A regular hexagon has six equal sides and six equal 120° angles. Draw lines from its center to the six corners, and you get six matching triangles with all sides equal (equilateral triangles). Thanks to that simple pattern, one measurement fixes all the others. That includes the side, the distance around, the area, the two kinds of diagonal, and the distances from the center to a side and to a corner.

Choose which measurement you have and enter it. The calculator turns it into the side length first, then works out the rest. It's handy for hexagonal tiles, pavers, nuts and bolt heads, honeycomb panels, garden beds and board-game grids.

Where it's used

Why this math matters outside the classroom:

  • Patio pavers. A hexagonal paver with 10 cm sides covers (3√3 ÷ 2) × 10², about 259.8 cm², of patio.
  • Nuts and bolts. A hex nut with 10 mm sides needs a wrench of about 10√3 ≈ 17.3 mm, its width across the flat sides.
  • Honeycomb. Bees build cells as near-regular hexagons, which tile a surface with no gaps while using the least wall for the area enclosed.
  • Game tiles. Board game designers who print hexagon tiles 2 in on a side allow 4 in from each corner to the opposite corner.
  • Gazebo floors. A builder framing a hexagonal gazebo with 6 ft sides needs 36 ft of perimeter beam and covers about 93.5 square feet.

Grade level and standards: Grades 7–8; Common Core State Standards for Mathematics 7.G.B.6, 8.G.B.7. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Select what you know: side, perimeter, area, short diagonal, long diagonal, inradius or circumradius.
  2. Enter the value. Any length unit works. If you enter an area, use the matching square unit.
  3. Read the other measurements. The headline is the area, or the side length when you started from the area.

What the inputs mean

Area
In square units of the same length unit, for example cm² for a side in cm.
Short diagonal
A diagonal that skips one corner. It equals the distance across the flats (from one side to the opposite side). That's the size used for a wrench on a hexagonal nut.
Long diagonal
The diagonal through the center, from one corner to the opposite corner: the distance across the corners. It's exactly twice the side.
Inradius (apothem)
Also called the apothem: the distance from the center to the middle of a side. It's the radius of the largest circle that fits inside.
Circumradius
The distance from the center to a corner. It's the radius of the circle passing through all six corners. In a regular hexagon it equals the side.

Formula and method

Perimeter: P = 6a
Area: A = (3√3 ÷ 2) × a² ≈ 2.598076 × a²
Short diagonal: d = √3 × a
Long diagonal: D = 2a
Inradius (apothem): r = (√3 ÷ 2) × a
Circumradius: R = a

In these formulas, a is the side length, P the perimeter and A the area. d is the short diagonal and D the long one. r is the inradius (center to the middle of a side) and R the circumradius (center to a corner). √3 is the square root of 3, about 1.732.

Each of the six triangles around the center is equilateral with side a. So the distance from the center to any corner is a. The long diagonal passes through two of those, so it's 2a. The height of one triangle, (√3 ÷ 2)a, is the inradius. Twice that is the distance across the flats, which equals the short diagonal √3 × a.

Each equilateral triangle has an area of (√3 ÷ 4)a², and six of them give (3√3 ÷ 2)a². Here's another way to see it: the area is half the perimeter times the inradius, ½ × 6a × (√3 ÷ 2)a.

To work backwards, the calculator rearranges the matching formula to find a. For example, from the area it uses a = √(2A ÷ 3√3). It keeps your own value exactly as entered.

Worked example

A hexagonal floor tile has sides of 10 cm. How much floor does it cover, and how wide is it?

  1. Perimeter: 6 × 10 = 60 cm.
  2. Area: (3√3 ÷ 2) × 10² ≈ 259.8076 cm².
  3. Width across the flats (short diagonal): √3 × 10 ≈ 17.3205 cm.
  4. Width across the corners (long diagonal): 2 × 10 = 20 cm.
  5. Inradius: (√3 ÷ 2) × 10 ≈ 8.6603 cm. Circumradius: 10 cm.

Result: Each tile covers about 259.8076 cm². So about 38.5 tiles fill a square meter, before allowing for grout and cuts.

Understanding the result

The two widths differ: across the flats is about 86.6% of the width across the corners. Check which one a product listing gives before you order.

Hexagons cover a flat surface with no gaps. Only three regular shapes can do that: triangles, squares and hexagons. Of the three, the hexagon holds the most area for a given perimeter.

Assumptions and limitations

  • The hexagon is regular: all sides equal and all angles 120°.
  • All lengths use the same unit, and the area is in that unit squared.
  • Irregular hexagons, with unequal sides or angles, need to be split into triangles and measured separately.
  • No unit conversion is built in. Use the length or area converter if you need a different unit.

Frequently asked questions

How do I find the area of a hexagon from the width across the flats?

Choose short diagonal and enter the flat-to-flat width. The side is that width divided by √3, and the area follows from the side. A 30 mm flat-to-flat hexagon has sides of about 17.32 mm and an area of about 779.42 mm².

Why is the circumradius equal to the side?

Joining the center to the six corners makes six triangles, each with a 60° angle at the center (360° ÷ 6). Each has two equal sides with a 60° angle between them, which makes it equilateral. So the center-to-corner distance equals the side.

What is the apothem used for?

The apothem (inradius) gives a quick area formula for any regular polygon: area = ½ × perimeter × apothem. For a hexagon with side 10, that's ½ × 60 × 8.6603 ≈ 259.81.

How many hexagons fit in a given area?

Divide the area to cover by the area of one hexagon. Hexagons fit together perfectly, so there's no wasted space between them, only at the edges of the region.

How this calculator is checked

Method: Regular hexagon decomposed into six equilateral triangles; standard relations for perimeter, area, diagonals, inradius and circumradius. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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