Year 6 Maths 3D Objects, Volume & Capacity

Cross-sections of prisms

20 practice questions 0 video lessons Theory + worked examples

Every question with a fully worked solution.

Theory

A cross-section is the flat face you see when a solid is cut straight through, like the face of a slice of bread. Cut a prism parallel to its ends and every slice is the same shape and size as the ends. Slices of pyramids, cones and spheres change size. This Year 6 maths page shows how to picture cross-sections, how the direction of the cut matters, and how to name a prism from its cross-section.

A cross-section (or slice) is the shape of the flat face made by one straight cut through a solid.

A parallel cross-section comes from a cut parallel to the base (or to the ends). For a prism, every parallel cross-section is the same shape and size as the ends: that is what makes it a prism.

For a pyramid or a cone, parallel slices keep the shape of the base but get smaller towards the apex. Every slice of a sphere is a circle, biggest through the centre.

Cutting in a different direction gives a different shape: a cylinder cut across gives a circle, but cut straight down through its centre it gives a rectangle.

A prism is named after its cross-section: a hexagon cross-section means a hexagonal prism.

A slice parallel to the ends of a hexagonal prism is the same hexagon.

Pyramid slices parallel to the base are squares that get smaller towards the apex.

A cylinder cut parallel to its base: a circle the same size as the base.

Cross-sections parallel to the base or ends:

Solid	Slice shape	Size along the solid
Prism	same as the ends	always the same
Cylinder	circle	always the same
Pyramid	same shape as the base	smaller near the apex
Cone	circle	smaller near the tip
Sphere	circle	biggest at the centre

          Same slice all along = prism-like. If every parallel slice matches, the solid has a constant cross-section (prisms and cylinders).
        

          Direction matters. The same solid can give different shapes when cut in different directions.
        

Finding a cross-section

Picture the cut and the flat face it leaves.
Parallel to the ends of a prism? The face is the end shape, the same size.
Another direction? Look at which faces the cut passes through: each flat face it crosses gives one side of the new shape.

Naming a prism from its cross-section

Count the sides of the cross-section.
Name the prism after that shape (triangular, pentagonal, hexagonal ...).

Example 1 — A pencil

An unsharpened pencil is a hexagonal prism. What shape is a slice cut straight across it, and does it matter where you cut?

Solution

A hexagon. Every slice across is the same size, because a prism has the same cross-section all along.

Example 2 — A wooden block

A block is \(10\) cm long, \(6\) cm wide and \(3\) cm high. It is cut parallel to its small \(6\) cm by \(3\) cm end. Describe the cut face.

Solution

A rectangle \(6\) cm by \(3\) cm, the same as the end.

Example 3 — Cut another way

The same block is cut parallel to its top, halfway up. Describe the cut face.

Solution

A rectangle \(10\) cm by \(6\) cm, the same as the top.

Example 4 — Cylinder two ways

A cylinder is cut
(i) across, parallel to the base
(ii) straight down through the centre of the top.
What shapes are the cut faces?

Solution

(i) A circle
(ii) A rectangle

Example 5 — A cone

Slices are cut from a cone, parallel to its base, moving up towards the tip. How do they change?

Solution

They are all circles, getting smaller towards the tip. So a cone does not have the same cross-section all along.

Example 6 — Halfway up a pyramid

A square pyramid has a base with sides of \(8\) cm. It is cut parallel to the base, halfway up. What is the cut face?

Solution

\(8 \div 2\)

\(=\)

\(4\)

8 \div 2 = 4

A square with sides of \(4\) cm: same shape as the base, but smaller.

Example 7 — Name the prism

A prism has a pentagon as its cross-section.
(i) Name the prism.
(ii) How many faces does it have?

Solution

\(5 + 2\)

\(=\)

\(7\)

5 + 2 = 7

(i) A pentagonal prism
(ii) \(7\) faces (\(5\) rectangles and \(2\) pentagons)

Example 8 — Which keep the same slice?

Which of these have the same cross-section all along: a can of soup, a rectangular eraser, an ice-cream cone, a tennis ball, a hexagonal pencil?

Solution

The can, the eraser and the pencil. The cone and the ball give slices that change size.

Example 9 — Other cuts

What shape is the cut face when
(i) a square pyramid is cut straight down through its apex and the middle of two opposite base edges
(ii) a cylinder is cut on a slant that does not touch the top or the bottom?

Solution

(i) A triangle
(ii) An oval (a stretched circle)

Example 10 — Find the mistake

Amir says a cylinder cannot have a cross-section because it has a curved surface. What is right?

Solution

A straight cut across a cylinder gives a flat circle. Like a prism, it has the same cross-section all along.

Common pitfalls

Thinking every slice of a pyramid is the same. Pyramid slices shrink towards the apex.

Ignoring the direction of the cut. A cylinder can give a circle or a rectangle.

Naming a prism by the face it rests on. Use the cross-section (the end shape).

Drawing the outside instead of the cut face. The cross-section is the flat face left by the cut.

Frequently asked questions

What is a cross-section?

The flat face made by a straight cut through a solid.

Why are all the slices of a prism the same?

A prism has the same cross-section all along its length.

What shape is a slice of a cylinder?

A circle if cut across; a rectangle if cut straight down through the centre.

What happens to slices of a pyramid?

They keep the base shape but get smaller towards the apex.

How do I name a prism from its cross-section?

Name it after the shape: a hexagon cross-section makes a hexagonal prism.

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Cross-sections of prisms

📖 Theory

Finding a cross-section

Naming a prism from its cross-section

Common pitfalls

Frequently asked questions

Theory