doodleshapes

Fact sheet · updated

SHAPES, IN PLAIN ENGLISH

Count the
corners.

Every solid shape has faces, edges and vertices.

Try a shape
Wooden geometric solids, a cube, a pyramid, a cylinder and a sphere, arranged on a light table by a window with soft daylight
AI-generated image, not a photograph of a real person or place.

V − E + Falways 2 for a polyhedron

In England, children start naming 2D and 3D shapes in Year 1 and are counting edges, vertices and faces by Year 2. Count all three on almost any box-like solid and they fit one tidy rule.

At a glance

5 solids

Solids made of identical regular faces

The tetrahedron, cube, octahedron, dodecahedron and icosahedron. Their faces number 4, 6, 8, 12 and 20.

PLATONIC SOLIDS · PLUS MAGAZINE

≈ 40,000 columns

Interlocking basalt columns

Nearly 40,000 columns, some as tall as 36 feet, formed as lava from an eruption around 60 million years ago cooled, shrank and cracked.

GIANT'S CAUSEWAY · APS

TWO NUMBERS WORTH KNOWING

Few perfect solids,
many near-hexagons.

One number from geometry,
one from a coastline in Northern Ireland.

Research reported by the American Physical Society found the columns are not quite as perfectly hexagonal as once believed: the crack network evolves into an almost hexagonal pattern.

A QUICK SHAPE CHECK

Pick a solid,
test the rule.

Choose a shape to see its vertices (V), edges (E) and faces (F). The bar shows how many faces it has to count.

Euler's formula says V − E + F = 2 for these solids. A shape with a hole through it follows a different total.

Shape check

Choose a shape
Tetrahedron4 − 6 + 4 = 2

4 vertices, 6 edges and 4 triangular faces. It has the fewest faces of the five Platonic solids.

Cube8 − 12 + 6 = 2

8 vertices, 12 edges and 6 square faces. A good one to count on a real box or a dice.

Icosahedron12 − 30 + 20 = 2

12 vertices, 30 edges and 20 triangular faces. More parts to count, and the same answer.

Doughnut shapeV − E + F = 0

A polyhedron with one hole through it does not give 2. The rule becomes 2 minus 2 for every hole, so one hole gives 0.

Sources: Plus magazine, University of Cambridge (cube, icosahedron, holes); ProofWiki (tetrahedron).

THE SHORT ANSWER

Three words do
most of the work.

Faces, edges and vertices.
Learn them once, use them on every solid.

FACES

The flat surfaces

A polyhedron is a solid whose surface is made of flat faces. A cube has 6 of them, each one a square.

Try a shape

EDGES AND VERTICES

Sides and corners

The sides of the faces are the edges; the corners are the vertices. Year 2 pupils in England learn to count both.

Count at home

IN NATURE

Hexagons everywhere

Honeycomb cells and the basalt columns of the Giant's Causeway both come out close to hexagonal.

Why hexagons
A close view of a honeycomb frame with neat hexagonal wax cells, some capped with honey, lit by natural light
AI-generated image, not a photograph of a real person or place.

FOUR THINGS TO TRY

Shapes,
around the house.

None of these needs special kit. Boxes, tins, dice and balls cover most of the 3D shapes children meet in primary school.

  1. Go on a shape hunt

    Find a cuboid, a cylinder and a sphere in the kitchen. Naming common 2D and 3D shapes is Year 1 work.

  2. Count on a real box

    Mark each corner with a sticker as you count, so no vertex is counted twice. Then count edges and faces.

  3. Check the rule

    Work out vertices minus edges plus faces. For a box, 8 − 12 + 6 gives 2.

  4. Flatten it into a net

    Open a cereal box out flat to see the 2D shapes it is made from. Making nets is in the Year 6 curriculum.

WHY HEXAGONS

The shortest
way to share.

If you split a flat surface into equal areas, the regular hexagon pattern of a honeycomb uses the least total perimeter. Pappus discussed the problem, and it was long known as the honeycomb conjecture.

Mathematician Thomas Hales set out the first general proof in 1999. On the Giant's Causeway, cooling lava cracked into a pattern that also tends towards hexagons.

An adult's hand pointing to a corner of a wooden cube while a small hand traces one of its edges, seen from above
AI-generated image, not a photograph of a real person or place.

Questions

Questions people ask

What is the difference between a vertex and an edge?

On a solid made of flat faces, the edges are the sides of those faces and the vertices are the corners where they meet. A cube has 12 edges and 8 vertices.

What is Euler's formula?

For a polyhedron, the number of vertices minus edges plus faces equals 2. It is named after the Swiss mathematician Leonhard Euler (1707 to 1783), though he was not the first to give a complete proof.

When do children learn about faces, edges and vertices?

In England's national curriculum, pupils name common 2D and 3D shapes in Year 1 and describe 3D shapes by their edges, vertices and faces in Year 2.

What are the Platonic solids?

The five solids whose faces are identical regular shapes: the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

Why are honeycomb cells hexagons?

A regular hexagon grid divides a surface into equal areas using the least perimeter, an idea known as the honeycomb conjecture. Thomas Hales gave the first general proof in 1999.

Sources

Where the facts come from

  1. GOV.UK: National curriculum in England, mathematics programmes of study
  2. Plus magazine, University of Cambridge: Euler's polyhedron formula
  3. ProofWiki: Euler Polyhedron Formula, examples
  4. Thomas C. Hales: The Honeycomb Conjecture (arXiv, 1999)
  5. American Physical Society: The Giant's Causeway

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