GCSE · Maths · AQA · Spec 8300

3D shapes — properties of faces, surfaces, edges, vertices

A sphere has no edges or corners, a cylinder has edges but no corners, and a cone has one of everything. Decide where each solid belongs.

Sort the solids

Seven solids, three questions

Use these meanings. A face is a flat surface; a curved surface is one that is not flat. An edge is a line where two surfaces meet, and it can be straight or curved. A vertex is a corner point where edges meet, or a point that a surface narrows to, like the tip of a cone. Place each solid: it can go in one set, in several, or in none.

  • A Has a curved surface
  • B Has at least one edge
  • C Has at least one vertex
  1. Cylinder
  2. Cone
  3. Sphere
  4. Cube
  5. Cuboid
  6. Prism (triangular, hexagonal or any other polygon end)
  7. Pyramid (any polygon base)

Predict, then check

A drawing of a solid hides the edges at the back. Count what the solid has, not what the picture shows.

A pentagonal prism has two identical pentagon ends joined by rectangles. How many edges does it have?

Now try a pyramid

This solid has one base and a single apex instead of two ends.

A square-based pyramid is a square base with four triangles meeting at the apex. How many edges does it have?

Families of solids

How the seven solids are related

Tick every property each solid has. A cross-section is the shape you get by slicing straight across the solid, parallel to its end or base.

Cube
Cuboid
Prism (e.g. triangular or hexagonal)
Cylinder
Pyramid (e.g. square-based)
Cone
Sphere

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Seven solids, four words, and the curved solids show what those words really mean.

What you need to know

  • The four words: face (a flat surface), curved surface, edge (where two surfaces meet) and vertex (a corner, or the point a surface narrows to).
  • The seven solids: cube, cuboid, prism, cylinder, pyramid, cone and sphere.
  • Counts come from how a solid is built, not from the lines a drawing happens to show.

The big picture

Every solid on the list is described with the same four words: face, curved surface, edge and vertex. Apply the definitions exactly and the curved solids stop looking like exceptions. Count a flat-faced solid from how it is built (two ends joined by rectangles for a prism, a base joined to one apex for a pyramid), not from the lines a drawing shows.

Key points

1A face is flat and a curved surface is not. Of the seven solids, only the cylinder, cone and sphere have a curved surface.
2An edge can be curved: a cylinder has 2 curved edges, a cone has 1, and a sphere has none because it has only one surface.
3A cone's apex is its 1 vertex. A cylinder and a sphere have no vertex.
4A prism whose ends have n sides has n + 2 faces, 3n edges and 2n vertices; a pyramid whose base has n sides has n + 1 faces, 2n edges and n + 1 vertices.
5A cube is a special cuboid and a cuboid is a prism. A cylinder shares the prism's constant cross-section, and a cone comes to an apex like a pyramid.

Worked example

Problem

A prism has 18 edges. How many sides does each of its ends have, and how many faces and vertices does it have?

⚠ Watch out

Counting only what a drawing shows. Edges and vertices at the back are hidden but still count, and the outline of a curved solid is not an edge: the circle drawn for a sphere and the two slanting lines drawn for a cone are outlines only.

🧠

Memory hook

Sphere: one surface, nothing else. Cylinder: edges, but no corners. Cone: one of everything (1 flat face, 1 curved surface, 1 edge, 1 vertex).

✓

Check yourself

Without drawing it, work out how many faces, edges and vertices an octagonal prism has (its two ends each have 8 sides). Then say which of its faces are rectangles.

Flashcards

(7)
What is the difference between a face and a curved surface?
A face is flat. A curved surface is not flat. The cylinder, cone and sphere each have one curved surface.
What is an edge, and can an edge be curved?
An edge is a line where two surfaces meet. Yes: where a cylinder's circular ends meet its curved surface, the edges are circles.
What counts as a vertex?
A corner point where edges meet, or a point that a surface narrows to, such as the apex of a cone.
Why does a cone have a vertex when a cylinder has none?
A cone's curved surface narrows to a single point, its apex. A cylinder's curved surface never comes to a point; it only meets the two circular faces along edges.
Why does a sphere have no edges?
It has only one surface, so there are never two surfaces meeting. The circle in a drawing of a sphere is its outline, not an edge.
How is a prism built, and what does that give for an n-sided end?
Two identical polygon ends joined by rectangles, one per side. That gives n + 2 faces, 3n edges and 2n vertices.
How do the cube, cuboid and prism fit together?
A cube is a cuboid whose faces are all squares, and a cuboid is a prism whose cross-section is a rectangle.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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