Naming and Properties of 3D Solids

Name a 3D solid from its picture, tell a prism from a pyramid, and count its faces, edges and vertices in groups so none are missed.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580 · 21 real past-paper questions below

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Faces, Edges and Vertices

A solid has three kinds of part. You need all three words in the exam.

Part	What it is
Face	a flat or curved surface
Edge	a line where two faces meet
Vertex	a corner where edges meet (plural: vertices)

A cube has 6 faces, 12 edges and 8 vertices.

Common mistake

Mixing up edges and vertices. Edges are lines you could run a finger along. Vertices are points you could touch with one fingertip.

Prism or Pyramid? Name the Solid

The 1-mark question: write down the mathematical name of this solid. Ask two questions, in this order.

Steps

1
Is it the same shape all the way through, like a loaf of bread? Then it is a prism. Name it by its end: triangular prism, pentagonal prism.
2
Does it rise from a flat base to one point? Then it is a pyramid. Name it by its base: square-based pyramid.
3
Six rectangles is a cuboid. Six squares is a cube. Both are prisms with special names.

Common mistake

Giving a flat name to a solid. A cuboid is not a rectangle and a cube is not a square. Rectangles and squares are its faces.

Key idea

A pyramid on a triangle base, with 4 triangle faces in all, is also called a tetrahedron.

Count a Prism in Groups

The 1-mark question: how many faces, edges or vertices has the prism? The drawing hides some at the back. So count in groups, not by pointing.

Start by counting the sides of one end. Call it the end count.

Worked example

1
$L shaped end: 6 sides$ — the end count. Count round the L slowly.
2
$faces = 6 + 2 = 8$ — one rectangle per side, plus the 2 ends
3
$edges = 6 + 6 + 6$ — round the front, round the back, and the ones joining them
4
$= 18$ — three groups of the end count
5
$vertices = 6 + 6 = 12$ — every corner sits on one of the 2 ends

Common mistake

Counting only the lines you can see. Hidden edges are drawn dashed, or not drawn at all. The groups count them for you.

Key idea

Check: faces + vertices − edges = 2. Here 8 + 12 − 18 = 2. Leonhard Euler wrote down this rule in a letter in 1750, and it works for every prism and pyramid.

Count a Pyramid: Base, Then Tip

The 2-mark question: complete the statement, the pyramid has … edges and … vertices. Count round the base first, then up to the tip.

Worked example

1
$pentagon base: 5 sides$ — the base count
2
$edges = 5 + 5 = 10$ — 5 round the base, 5 up to the tip
3
$vertices = 5 + 1 = 6$ — 5 base corners, plus the tip
4
$faces = 5 + 1 = 6$ — 5 triangles, plus the base

Common mistake

Forgetting the tip, or forgetting the base. The tip is a vertex and the base is a face.

Key idea

Check: 6 + 6 − 10 = 2. Correct.

Curved Solids: Cylinder, Cone, Sphere

The 1-mark question often shows a round solid. Name it from its flat faces.

Solid	What it has
Cylinder	2 flat circles, 1 curved surface, 2 curved edges, no vertices
Cone	1 flat circle, 1 curved surface, 1 curved edge, 1 vertex at the tip
Sphere	1 curved surface, no edges, no vertices
Hemisphere	half a sphere: 1 flat circle, 1 curved surface, 1 curved edge

Common mistake

Calling a cone a pyramid because it comes to a point. A pyramid has flat triangle faces and a base with straight sides. A cone has a round base and a curved side.

Which Method? Read the Question

Name the solid first. The counts follow from the end or the base.

You see	You do
Name this solid	same all through: prism; up to a point: pyramid
Faces of a prism	end count + 2
Edges of a prism	front + back + joining
Vertices of a prism	end count, twice
Edges of a pyramid	base count, twice
Vertices of a pyramid	base count + 1
Round solid	count flat circles: 2 cylinder, 1 cone, 0 sphere
Net, or wire for the edges	Nets of 3D Shapes
Planes of symmetry	Planes of Symmetry in 3D Shapes

Key idea

For a prism or pyramid, finish with faces + vertices − edges = 2. If it is not 2, recount.

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