3D objects - Year 7 Mathematics

Year 7 core

By the end of this topic you should be able to:

name a prism or pyramid from the shape of its base,
count the faces, edges, and vertices of simple 3D objects,
recognise and draw nets of cubes, rectangular prisms, triangular prisms and square pyramids,
interpret isometric and perspective drawings of solids.

1. Prisms and pyramids

A triangular prism (two triangular ends joined by rectangles) and a square pyramid (a square base with triangular side-faces meeting at an apex).

A prism has two identical, parallel end-faces (the bases) joined by rectangular side-faces. It is named after the shape of its base.

Prism	Base shape
Triangular prism	triangle
Rectangular prism (cuboid)	rectangle
Pentagonal prism	pentagon
Cube	square

A pyramid has a polygon base and triangular side-faces that meet at a single point (the apex).

Pyramid	Base shape
Triangular pyramid (tetrahedron)	triangle
Square pyramid	square
Pentagonal pyramid	pentagon

2. Faces, edges, vertices

Face - a flat surface of the solid.
Edge - a line where two faces meet.
Vertex (plural: vertices) - a corner where edges meet.

Solid	Faces	Edges	Vertices
Cube	$6$	$12$	$8$
Rectangular prism	$6$	$12$	$8$
Triangular prism	$5$	$9$	$6$
Square pyramid	$5$	$8$	$5$
Triangular pyramid (tetrahedron)	$4$	$6$	$4$

3. Nets

A net is a flat pattern that folds up to form a 3D shape. There are $11$ different nets for a cube (all made from six squares arranged so they fold into a closed box).

A net of a cube: six squares laid flat that fold up into a cube. Eleven different nets work; this is one.

Worked example 1 Triangular prism net

A triangular prism has $2$ triangular end-faces and $3$ rectangular side-faces. Its net has:

$2$ triangles matching the two ends,
$3$ rectangles whose widths match the sides of the triangles.

Laid out in order, the three rectangles form a single long strip with the two triangles attached to one side.

4. Two-dimensional representations

Solid objects can be drawn in 2D in several useful ways:

Net - a shape that folds into the solid. Useful for building, cutting, wrapping.
Isometric drawing - made on triangular grid paper; all three axes shown at $60 deg$ . Lengths along the axes are actual sizes. Useful for engineering sketches.
Perspective drawing - farther things look smaller; converges to a vanishing point. Closer to how we see objects in real life.
Top / side / front views - three orthogonal views on square grid paper. Used in architectural plans.

Practice: Year 7 core

Fluency

Naming, faces, edges, vertices

How many faces, edges, and vertices does a cube have?
How many faces, edges, and vertices does a triangular prism have?
How many faces, edges, and vertices does a square pyramid have?
Name the solid with $4$ triangular faces and $4$ vertices.
Name the solid with a hexagonal base and $6$ rectangular side-faces.
How many rectangular faces does a pentagonal prism have?
How many triangular faces does a square pyramid have?

Fluency

Nets

How many different nets fold into a cube?
List the shapes in a net of a triangular prism.
List the shapes in a net of a rectangular prism with dimensions $3 \times 2 \times 4$ .
Draw (or describe) a net for a square pyramid with base side $4$ cm and slant height $5$ cm.
A net of six squares in one long straight row - can this fold into a cube? Explain.

Reasoning

Explain and spot the mistake

Maya says: “every pyramid has a triangular base”. Is this true? Explain.
Sam counts the faces of a triangular prism as $6$ . Where could Sam’s error be?
Is a cylinder a prism? Explain using the definition of a prism.
You see the isometric drawing of an object but cannot tell whether the left side is longer than the front side. Describe one extra drawing that would resolve the ambiguity.

Problem solving

Real-world problems

A cereal box is a rectangular prism $30 \times 20 \times 8$ cm. Find its total surface area and its volume.
A Toblerone box is a triangular prism with equilateral cross-section (side $6$ cm) and length $24$ cm. Describe the net (number and size of each shape).
A square pyramid has base $8$ cm and slant height $5$ cm. Find its total surface area. (Base area $+ 4 \times$ triangular side area.)
A shipping crate is a $1$ m cube. What is the total length of all its edges?

Extension

Counting formulas and Euler

Prism with an n-sided base

V = 2n, \quad E = 3n, \quad F = n + 2.

A triangular prism ( $n = 3$ ) has $V = 6$ , $E = 9$ , $F = 5$ .

Pyramid with an n-sided base

V = n + 1, \quad E = 2n, \quad F = n + 1.

A square pyramid ( $n = 4$ ) has $V = 5$ , $E = 8$ , $F = 5$ .

Euler's formula (every convex polyhedron)

V - E + F = 2.

Practice: Extension

Reasoning

Using the formulas

A prism has a $7$ -sided base. Find $V$ , $E$ , $F$ .
A pyramid has a hexagonal base. Find $V$ , $E$ , $F$ .
A solid has $V = 20$ , $E = 30$ . Use Euler’s formula to find $F$ .
A solid has $V = 6$ , $F = 8$ . Find $E$ .
A solid has $8$ triangular faces. Give its name.

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

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Year 7 core - answers

Fluency

Naming, faces, edges, vertices

Cube: $F = 6$ , $E = 12$ , $V = 8$ .
Triangular prism: $F = 5$ , $E = 9$ , $V = 6$ .
Square pyramid: $F = 5$ , $E = 8$ , $V = 5$ .
Triangular pyramid (tetrahedron).
Hexagonal prism.
$5$ rectangular faces (plus $2$ pentagonal ends).
$4$ triangular faces.

Fluency

Nets

$11$ .
$2$ triangles and $3$ rectangles.
$6$ rectangles: two $3 \times 2$ , two $3 \times 4$ , two $2 \times 4$ .
A $4$ cm square with four isosceles triangles attached to each side, each triangle with base $4$ cm and slant height $5$ cm.
No. Six squares in a single row overlap when folded - they cannot form a closed cube.

Reasoning

Explain and spot the mistake

Not true. A pyramid’s side-faces are always triangles, but the base can be any polygon (square, pentagon, etc.). “Triangular pyramid” is one particular type.
A triangular prism has $5$ faces: $2$ triangular ends and $3$ rectangular sides. Sam probably mixed faces with edges (which total $9$ ) or counted the same face twice.
Strictly no - a prism has a polygon base joined by flat rectangular sides. A cylinder has a circular base and a curved surface, not a polygon. (It is often informally called a circular prism because the volume formula $V = \text{base area} \times \text{height}$ still applies.)
A top view (plan) or a side elevation would resolve the ambiguity, since those show lengths directly without the isometric distortion.

Problem solving

Real-world problems

Surface area $2000$ cm^2; volume $4800$ cm^3. Method: SA $= 2(30 \times 20 + 30 \times 8 + 20 \times 8) = 2(600 + 240 + 160) = 2000$ ; $V = 30 \times 20 \times 8$ .
$2$ equilateral triangles (side $6$ cm) and $3$ rectangles ( $6$ cm by $24$ cm).
$144$ cm^2. Method: base $8^2 = 64$ ; four triangles each $\tfrac{1}{2} \times 8 \times 5 = 20$ ; total $= 64 + 4 \times 20$ .
$12$ m. Method: a cube has $12$ edges, each $1$ m.

Extension - answers

Reasoning

Using the formulas

$V = 14$ , $E = 21$ , $F = 9$ . Method: $n = 7$ ; $V = 2n$ , $E = 3n$ , $F = n + 2$ .
$V = 7$ , $E = 12$ , $F = 7$ . Method: $n = 6$ ; $V = n + 1$ , $E = 2n$ , $F = n + 1$ .
$F = 12$ . Method: $V - E + F = 2$ .
$E = 12$ . Method: $6 - E + 8 = 2$ , so $E = 12$ .
Regular octahedron.

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