How many faces, edges and vertices does a cube have?

A cube has 6 faces, 12 edges and 8 vertices. Every face is a square and every vertex is the meeting point of exactly three edges. A cuboid has the same counts, because it is a cube that has been stretched, which changes the sizes but not the structure.

What is Euler's formula for 3D shapes?

Vertices minus edges plus faces equals 2, usually written V - E + F = 2. Leonhard Euler described it in the 1750s. For a cube, 8 - 12 + 6 = 2. It works for any convex polyhedron, which makes it a quick way to check whether a child has miscounted.

How many faces does a cylinder have?

Most UK primary schools teach that a cylinder has 2 flat faces, 1 curved surface, 2 edges and 0 vertices. Some schemes count the curved surface as a third face and say 3 faces. Both answers appear in textbooks, so the practical advice is to use the wording your child's school uses and know that mathematicians do not call a cylinder a polyhedron at all.

Is a vertex the same as a corner?

In everyday classroom language, yes. Corner is the informal word and vertex is the mathematical one. Teachers usually introduce corner first and then swap in vertex, because vertex is the word that appears in test papers and in later geometry.

When do children learn about faces, edges and vertices?

In England, the national curriculum introduces the number of edges, vertices and faces of 3D shapes in Year 2, when children are aged six to seven. The vocabulary is then revisited through Key Stage 2 alongside work on nets, prisms, pyramids and volume.

What Are Vertices, Faces and Edges? A Guide for Parents and Teachers

Q: What are vertices, faces and edges?

A face is a flat surface on a 3D shape. An edge is the straight line where two faces meet. A vertex is the point where edges meet, which children often call a corner. The plural of vertex is vertices, and the singular is vertex rather than vertice.

2 hours ago

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Hand a seven-year-old a cube and ask how many faces it has. A good number will say three, and they are not being careless. From any angle, three faces is exactly what you can see. The other three are round the back, and the whole topic turns on getting children to count what is there rather than what is visible.

Key Takeaways

● A face is a flat surface, an edge is where two faces meet and a vertex is where edges meet.

● A cube has 6 faces, 12 edges and 8 vertices. So does a cuboid.

● Euler's formula, V - E + F = 2, holds for every convex polyhedron and catches miscounts in seconds.

● Prisms and pyramids follow rules. A prism on an n-sided base has 2n vertices, 3n edges and n+2 faces.

● Cylinders, cones and spheres are the awkward ones, because they have curved surfaces and are not polyhedra.

● In England this vocabulary first appears in Year 2.

Colorful classroom math scene with 3D shape models, a chalkboard of solids, and notebook labels Vertex, Edge, Face.

The three definitions, properly

Term	Definition	Everyday word	Watch out for
Face	A flat surface on a 3D shape	Side	"Side" also means edge in 2D work, which confuses children
Edge	The straight line where two faces meet	Line	Easy to miss the hidden ones at the back
Vertex	The point where two or more edges meet	Corner	The plural is vertices. The singular is vertex, never "vertice"

That last row causes more lost marks than anything else in the topic. "How many vertice does it have" is wrong, and once a child has written it a few times the habit sticks.

A quick way to make the three ideas physical: build shapes from cocktail sticks and marshmallows. The sticks are edges, the marshmallows are vertices and the gaps you can see through are where the faces would be. A child who has built a triangular prism with nine sticks and six marshmallows does not need to be told the numbers.

Faces, edges and vertices of common 3D shapes

Shape	Faces	Edges	Vertices	V - E + F
Cube	6	12	8	2
Cuboid	6	12	8	2
Triangular prism	5	9	6	2
Pentagonal prism	7	15	10	2
Hexagonal prism	8	18	12	2
Square-based pyramid	5	8	5	2
Triangular-based pyramid (tetrahedron)	4	6	4	2
Hexagonal pyramid	7	12	7	2
Octahedron	8	12	6	2
Cylinder	2 flat, 1 curved surface	2	0	does not apply
Cone	1 flat, 1 curved surface	1	1 apex	does not apply
Sphere	1 curved surface	0	0	does not apply

Notice that the cube and the cuboid are identical in that table. Stretching a cube changes the measurements and nothing else, which is a useful thing to point out, because children often assume a longer shape must have more of something.

The shortcuts worth teaching

Once a child can count reliably, give them the patterns. They turn a slow exercise into a five-second answer and they work for any base.

Prisms. A prism has two identical end faces joined by rectangles. For an n-sided base:

● Faces = n + 2

● Edges = 3n

● Vertices = 2n

Check it on a hexagonal prism. Six sides, so 8 faces, 18 edges, 12 vertices. That matches the table.

Pyramids. A pyramid has one base and triangular faces meeting at an apex. For an n-sided base:

● Faces = n + 1

● Edges = 2n

● Vertices = n + 1

A square-based pyramid gives 5, 8 and 5.

Euler's formula, and why it is the best marking tool you have

In the 1750s, the Swiss mathematician Leonhard Euler noticed that for any convex polyhedron:

V - E + F = 2

Vertices minus edges plus faces always lands on two. For a cube, 8 - 12 + 6 = 2. For a triangular prism, 6 - 9 + 5 = 2. For a tetrahedron, 4 - 6 + 4 = 2.

The practical use in a classroom is immediate. A child tells you a pentagonal prism has 7 faces, 15 edges and 12 vertices. Run the formula: 12 - 15 + 7 = 4. Something is wrong, and because faces are the easiest to count and vertices the easiest to double-count, the vertex figure is where you look first. It is a check children can run themselves, which is the part that matters.

Why cylinders and cones break the rules

This is where parents and children end up arguing with the homework, and both sides are usually right.

A polyhedron is a solid whose faces are all flat polygons. A cylinder is not one, because of the curved surface. Euler's formula applies to polyhedra, so it has nothing to say about cylinders, cones or spheres.

Schools still need children to describe these shapes, so conventions were invented. In most UK primary schemes a cylinder has 2 flat faces, 1 curved surface, 2 edges and 0 vertices. Some schemes say 3 faces, counting the curved surface as a face. For a cone, the usual line is 1 flat face, 1 curved surface, 1 edge and 1 vertex, though strictly the point at the top is an apex rather than a vertex, because no edges meet there.

There is no national answer to settle this. If a textbook and a teacher disagree, go with the teacher, because that is who marks the paper. It is also worth telling an older or curious child the truth: the question has no fixed answer because the shape sits outside the definition the vocabulary was built for.

Myths vs facts

Myth	Fact
A cube has more faces than a cuboid	Both have 6 faces, 12 edges and 8 vertices. Stretching changes size, not structure
"Vertice" is the singular of vertices	The singular is vertex. "Vertice" is not a word
A sphere has one face	It has one curved surface. A face is flat by definition, which is why most schemes avoid calling it a face
Euler's formula works for all 3D shapes	Only for convex polyhedra. It fails for curved solids and for shapes with holes, where a torus-like solid gives V - E + F = 0
There are lots of Platonic solids	There are exactly five: tetrahedron, cube, octahedron, dodecahedron and icosahedron. Euclid proved there can be no more
Counting is the only skill being tested	SATs-style questions usually give properties and ask for the shape, so children need to work backwards from the numbers too

Frequently asked questions

How do I stop my child miscounting hidden edges?

Make them touch each one and mark it. A whiteboard pen on a plastic shape, or small sticky dots, converts an abstract count into a physical task with visible evidence of what has already been counted.

What is the difference between a net and a 3D shape?

A net is the flat pattern that folds up into the solid. Nets are the fastest way to prove a face count, because every face has to appear on the net and can be counted lying flat.

Which 3D shape has 5 faces, 9 edges and 6 vertices?

A triangular prism. Questions phrased this way are common in end-of-key-stage papers, so practise reading the numbers back to a shape as well as forwards.

Do 2D shapes have faces and edges?

2D shapes have sides and vertices, not faces and edges. Keeping the two sets of words separate early saves a lot of confusion later.

What does Year 6 need beyond counting?

Nets, surface area, volume, naming prisms and pyramids by their base and recognising shapes from their properties. The vocabulary from Year 2 carries all the way through, which is why it is worth fixing properly at the start.

Is a cube a prism?

Yes. A cube is a square prism, and the prism rules give the right answer: four sides on the base, so 6 faces, 12 edges and 8 vertices.

Disclaimer: This guide follows the conventions used in the English national curriculum, and definitions for curved solids such as cylinders and cones vary between schemes and exam boards. Always check the wording your child's school uses before correcting their work.