Name Of Shape With Six Sides

You've seen it on a soccer ball. Practically speaking, you've walked across it in a bathroom tile pattern. That's why you've probably eaten it in honeycomb form, spread on toast. The hexagon is everywhere — and most people couldn't tell you why.

Six sides. Day to day, six angles. A shape that tiles a plane perfectly without gaps. It's the geometry of efficiency, and once you start noticing it, you can't unsee it.

What Is a Hexagon

A hexagon is a polygon with six straight sides and six interior angles. So the word comes from Greek — hex meaning six, gonia* meaning angle. Consider this: that's the textbook definition. But in practice, "hexagon" covers a few different things.

Regular vs. Irregular

A regular hexagon has six equal sides and six equal angles (each 120 degrees). Consider this: this is the one you picture when someone says the word. It's the honeycomb cell. Still, the nut on a bolt. The tiles in a strategy board game.

An irregular hexagon just needs six sides. A child's drawing of a house with a crooked roof? The side lengths can vary. The angles can be anything. Technically a hexagon if you count the outline.

Convex vs. Concave

Convex hexagons have all interior angles less than 180 degrees. No "dents.Plus, " Concave hexagons have at least one angle greater than 180 — a corner that caves inward. Think of a star shape with six points, or a hexagon with one side pushed in.

Both are still hexagons. The definition is surprisingly loose for something so geometrically precise.

Why It Matters / Why People Care

Nature didn't choose the hexagon by accident. That said, bees build honeycomb with hexagonal cells because it's the most efficient way to divide a surface into equal-area regions with the least total perimeter. Which means less wax. More storage. Evolution figured out calculus millions of years before humans did.

The Tiling Superpower

Only three regular polygons tile a plane by themselves: triangles, squares, and hexagons. Of those three, the hexagon has the smallest perimeter for a given area. That's why it shows up in:

• Basalt columns (Giant's Causeway, Devils Postpile) — cooling lava contracts into hexagonal fractures
• Dragonfly eyes — thousands of hexagonal lenses packed together
• Turtle shells — scutes often form hexagonal patterns
• Snowflakes — six-fold symmetry from molecular bonding angles

Engineers copy this. In real terms, the James Webb Space Telescope's primary mirror is 18 hexagonal segments. Satellite solar arrays. Modular flooring systems. The shape scales without waste.

Structural Strength

A hexagon distributes force evenly across its sides. That's why nuts and bolts are hexagonal — six contact points for a wrench, maximum torque transfer, minimum rounding risk. Day to day, square nuts strip easier. Octagonal nuts waste material and don't grip better.

How It Works

The Math You Actually Need

Interior angles: In a regular hexagon, each interior angle is 120°. Formula: (n-2) × 180° / n = (6-2) × 180° / 6 = 720° / 6 = 120°.

Exterior angles: Each is 60°. They always sum to 360° for any convex polygon.

Area of a regular hexagon: (3√3 / 2) × s², where s is side length. Roughly 2.598 × s².

Perimeter: 6s. Simple.

Radius relationships: The distance from center to vertex (circumradius) equals the side length. The distance from center to edge midpoint (inradius / apothem) is s × √3 / 2 ≈ 0.866s.

Constructing One With Compass and Straightedge

This is the classic Euclidean construction. This leads to draw a circle. Practically speaking, keep the compass at the same radius. Place the point on the circumference and mark an arc crossing the circle. In real terms, move the point to that intersection. Repeat. Six steps and you're back where you started. Connect the dots.

The circle's radius is the hexagon's side length. That's not a coincidence — it's why the construction works.

Coordinates for Digital Work

If you're coding a hex grid (games, data viz, procedural generation), center a regular hexagon at origin with flat tops:

Vertices (pointy-top orientation):
(0, R), (R√3/2, R/2), (R√3/2, -R/2), (0, -R), (-R√3/2, -R/2), (-R√3/2, R/2)

For flat-top orientation, swap x and y and adjust signs. The math is clean because 60° and 30° angles give you √3/2 and 1/2 factors — no messy decimals.

Hexagonal Grids vs. Square Grids

Square grids are easier to index (row, column). Hex grids have six neighbors instead of four (or eight with diagonals). Movement distance is more uniform — no "diagonal moves cost 1.414" problem.

Two common coordinate systems:

Axial coordinates (q, r) — two axes at 120°. Third coordinate s = -q - r is implicit. Neighbor offsets are constant.

Cube coordinates (x, y, z) — three axes, constraint x + y + z = 0. Elegant for distance calculations: distance = max(|dx|, |dy|, |dz|).

Want to learn more? We recommend why is 13 a bakers dozen and how long was the us in the vietnam war for further reading.

Most game devs pick axial for storage, convert to cube for math.

Common Mistakes / What Most People Get Wrong

Confusing Hexagon With Hexagram

A hexagram is a six-pointed star — two overlapping equilateral triangles. The Star of David. A hexagon is the convex hull of that star. They're related (the star's vertices are a hexagon's vertices) but they're not the same shape.

Assuming All Six-Sided Shapes Are "Hexagons" in the Useful Sense

Technically true. Practically useless. An irregular, concave hexagon with sides of wildly different lengths doesn't tile, doesn't distribute force evenly, and doesn't pack efficiently. The properties people care about — tiling, strength, efficiency — belong to the regular* hexagon specifically.

Thinking Hexagons Are Always "Better" Than Squares

They're not. Squares are easier to manufacture, easier to index in arrays, easier to cut from sheet goods with zero waste. Hexagons leave triangular gaps when cut from rectangles. Which means they complicate coordinate systems. They make rectangular enclosures awkward.

Use hexagons when their specific advantages matter. Default to squares otherwise.

Misremembering the Angle

People guess 60° for interior angles. Which means that's the exterior* angle. Interior is 120°. This matters if you're cutting wood, designing a tile pattern, or writing a shader.

Forgetting the √3 Factor

The height of a regular hexagon (flat-top) is 2 × apothem = s√3. Even so, the width (pointy-top) is also s√3. The √3 ≈ 1.

The √3 factor also appears in the hexagon’s area. For a regular hexagon with side length s, the area is

[ A = \frac{3\sqrt{3}}{2},s^{2}, ]

which follows from splitting the shape into six equilateral triangles of side s. The apothem (the distance from the center to the midpoint of a side) is

[ a = \frac{\sqrt{3}}{2}s, ]

so the perimeter is simply 6 s, and the “width” (flat‑top orientation) or “height” (pointy‑top orientation) equals 2a = s√3. Knowing these exact expressions lets you avoid floating‑point drift when you need to scale a hex grid to match a pixel‑perfect canvas or a physical layout.

Working with Axial Coordinates in Code

Most developers store hex positions as axial (q, r). Converting to cube coordinates for distance or rounding is straightforward:

def axial_to_cube(q, r):
    x = q
    z = r
    y = -x - z          # because x + y + z = 0
    return (x, y, z)

def cube_to_axial(x, y, z):
    return (x, z)       # q = x, r = z

Neighbor offsets in axial space are constant, which makes iteration cheap:

AXIAL_NEIGHBORS = [
    (+1,  0), (+1, -1), ( 0, -1),
    (-1,  0), (-1, +1), ( 0, +1)
]

def axial_neighbors(q, r):
    return [(q + dq, r + dr) for dq, dr in AXIAL_NEIGHBORS]

If you need to round a floating‑point cube coordinate back to the nearest hex (e.g., after a pixel‑to‑hex conversion), use the standard cube‑round algorithm:

def cube_round(x, y, z):
    rx, ry, rz = round(x), round(y), round(z)
    x_diff, y_diff, z_diff = abs(rx - x), abs(ry - y), abs(rz - z)
    if x_diff > y_diff and x_diff > z_diff:
        rx = -ry - rz
    elif y_diff > z_diff:
        ry = -rx - rz
    else:
        rz = -rx - ry
    return (rx, ry, rz)

Rendering Hexagons Efficiently

When drawing many hexes, pre‑compute the six vertex offsets for the chosen orientation and reuse them:

# pointy‑top, radius = distance from center to a vertex
R = size                     # side length
sqrt3_over_2 = 0.86602540378
vertices = [
    (0,               -R),
    ( sqrt3_over_2R, -R/2),
    ( sqrt3_over_2R,  R/2),
    (0,                R),
    (-sqrt3_over_2R,  R/2),
    (-sqrt3_over_2R, -R/2)
]

Translate each vertex by the hex’s center (cx, cy) and draw a filled polygon. Because the vertex list is static, the per‑hex cost is just a couple of additions and a polygon fill — ideal for real‑time games or interactive visualizations.

Why Hexagons Appear in Nature and Engineering

The regular hexagon is the solution to the planar “isoperimetric problem” for tiles: among all shapes that can tessellate the plane without gaps, it minimizes the perimeter for a given area. This is why honeybees build hexagonal wax cells — less wax, more storage. In materials science, graphene’s carbon lattice is hexagonal, giving it

unparalleled strength and stability. In engineering, hexagonal structures are frequently used in lightweight honeycomb sandwich panels, providing maximum rigidity with minimal weight.

Conclusion

Mastering hexagonal grids requires a shift in perspective from the standard Cartesian $(x, y)$ grid to the more symmetrical cube or axial coordinate systems. While the math—involving $\sqrt{3}$ and coordinate transformations—is slightly more complex than square-based grids, the benefits are substantial. Hexagons provide more uniform distances between neighbors, eliminate the "diagonal" ambiguity found in square grids, and offer a more natural, organic aesthetic for strategy games, map editors, and procedural terrain generation. By leveraging axial coordinates for logic and cube coordinates for geometric calculations, you can build highly performant and mathematically solid systems that scale effortlessly from simple prototypes to complex, large-scale simulations.