Naming 3D solids and their faces, edges and vertices, and matching solids to their nets.
Practise Geometry of 3D objects in the app →
What gets asked
- Name the number of faces, vertices or edges of a solid.
- Use V minus E plus F equals 2 to find a missing count.
- Identify which solid a given net will fold into.
- Describe the properties of a sphere or a cylinder.
You must be able to
- Count the faces, vertices and edges of a prism, pyramid or Platonic solid.
- Substitute known values into V - E + F = 2 and solve for the missing one.
- Picture a net folding up to check which solid it makes.
- Describe a cylinder's flat and curved surfaces, and a sphere's one curved surface.
Traps that cost marks
- Counting only the faces or edges that are visible in a drawing. Count every face and edge of the solid, including the ones hidden from view.
- Putting the counts into the wrong places in V - E + F = 2. Write down which count is V, E and F before you substitute.
- Choosing a net with the right number of faces but the wrong shapes. Check both the number of faces and the shape of each face against the solid.
Worked example
- Substitute: .
- Simplify: .
- vertices.
Platonic solids and nets
Faces, edges and corners: the five perfect solids, and the rule that ties their counts together.
- face
- A flat surface of a solid, such as one side of a matchbox.
- edge
- The straight line where two faces meet.
- vertex
- A corner, where edges meet. Two or more of them are called vertices.
- Platonic solid
- A solid whose faces are all the same regular shape, with the same number meeting at every vertex.
- polyhedron
- A solid with flat faces only. A ball is not one, because its surface is curved.
- net
- A flat pattern that folds up into a solid, with no gaps and no overlaps.
Count on the whole solid, not on the drawing. A cube on paper shows you three faces at most. It still has , and the hidden ones count.
There are exactly five Platonic solids. The tetrahedron has triangles, the cube has squares and the octahedron has triangles. The dodecahedron has pentagons and the icosahedron has triangles.
Every polyhedron obeys one rule: . counts vertices, counts edges and counts faces. Write down which is which before you substitute.
A sphere has no faces, no edges and no vertices. Its whole surface is curved. A cylinder has two flat circular ends joined by one curved surface, and a curved surface is not a face.
A net folds up with no gaps and no overlaps. Count the shapes in the net and check what each one is. Six squares can make a cube, but only if nothing overlaps when folded.
Rules to remember
- for any polyhedron:
Examples
Worked answer
- Write the counts down first: and .
- Substitute: .
- Simplify: .
- , so a cube has vertices.
Answer: vertices
Labelling each count before substituting keeps the faces out of the edge slot.
Worked answer
- Count the shapes: faces.
- A pyramid on a square base also has faces.
- But that pyramid needs triangles, and here there are .
- Two triangular ends joined by three rectangles make a triangular prism.
Answer: a triangular prism
The face count alone left two answers open. The face shapes chose between them.
Worked answer
- A sphere has one curved surface and nothing flat on it.
- So it has no faces, no edges and no vertices.
- That rule is only for a polyhedron, and a sphere is not one.
Answer: no faces, no edges and no vertices
A curved surface is not a flat face, so a sphere never enters the rule.
Traps
- Counting only the faces or edges that are visible in a drawing. Count every face and edge of the solid, including the ones hidden from view.
- Putting the counts into the wrong places in V - E + F = 2. Write down which count is V, E and F before you substitute.
- Calling a cylinder a prism and counting its curved surface as a face. A face is flat. A cylinder has two flat ends and one curved surface, so it is no polyhedron.
- Choosing a net with the right number of faces but the wrong shapes. Check both the number of faces and the shape of each face against the solid.