Cross-sections of Prisms (Year 8)

01 / LEARN

Understand the idea.

A shape carried through space

A prism has matching polygonal ends in parallel planes and the same end-shaped cross-section all the way along. This chapter uses right prisms, where the length direction is perpendicular to the ends. Parallel slices of a triangular prism are congruent triangles.

A cross-section is the flat region exposed by a plane cut, so its shape depends on the cutting direction. A cylinder cut parallel to its circular base gives a circle; a cut through its axis gives a rectangle.

Why cross-sections matter

To understand volume, imagine stacking thin copies of the repeated end shape. A larger slice area fills more space per layer; a longer solid contains more layers. For a right prism, $V = A L$ , using end area $A$ and perpendicular length $L$ .

Identify the repeated end-shaped slice before calculating its area. A triangle needs its own perpendicular base and height; the prism's length is a separate dimension. A pyramid's slices shrink towards the tip, so the unchanged prism rule does not apply.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Choose the right shape

A solid has two matching triangular ends and rectangular side faces. Name a cross-section parallel to an end.

Worked solution

1
Identify the repeated ends.
Two matching triangular ends joined by rectangular side faces identify a triangular prism.
2
Choose a cut parallel to an end.
The prism has the same end-shaped slice throughout its length.
3
State the shape and size of the section.
The cross-section is a triangle congruent to either triangular end, not merely any triangle.

Answer

A triangle congruent to an end face.

EXAMPLE 02

Calculate a cross-sectional area

A prism’s cross-section is a triangle with base 6 cm and perpendicular height 4 cm. Find its cross-sectional area.

Worked solution

1
Identify the triangle's two required dimensions.
$b = 6 cm, h = 4 cm$
The given height is perpendicular to the cross-section's base.
2
Apply the triangle area formula.
$A = \frac{1}{2} b h = \frac{1}{2} \times 6 \times 4$
A triangle occupies half the area of the matching base-height rectangle.
3
Calculate and label the area.
$A = 12 {cm}^{2}$
This measures the flat cross-section; no prism length is needed.

Answer

12 cm².

03 / YOUR TURN

Try it yourself.

Check your understanding

A rectangular prism has an end face 3 cm by 5 cm. What is the area of a parallel cross-section?

Show answer & reasoning

15 cm².

A section parallel to the end is congruent to the end rectangle. Its area is $3 \times 5 = 15 {cm}^{2}$ at every position along the prism.

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

Name a cross-section parallel to an end of a pentagonal prism.
1 mark
Name a cross-section of a cylinder cut parallel to its circular base.
1 mark
A prism has a rectangular end 7 cm by 4 cm. Find the area of a parallel cross-section.
2 marks
A triangular prism has end base 8 m and end perpendicular height 3 m. Find the end area.
2 marks
A student applies $cross-sectional area \times length$ to a pyramid. Explain the problem.
3 marks
A solid has a circular base and tapers to one point. Is it a prism or a cylinder? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Name a cross-section parallel to an end of a pentagonal prism.
1 mark
A prism repeats its end shape along its length.
A pentagon congruent to the end.
Name a cross-section of a cylinder cut parallel to its circular base.
1 mark
The cut has the same orientation as a circular end.
A circle.
A prism has a rectangular end 7 cm by 4 cm. Find the area of a parallel cross-section.
2 marks
$A = 7 (4) = 28$ cm².
28 cm².
A triangular prism has end base 8 m and end perpendicular height 3 m. Find the end area.
2 marks
$A = \frac{1}{2} (8) (3) = 12$ m².
12 m².
A student applies $cross-sectional area \times length$ to a pyramid. Explain the problem.
3 marks
The prism rule relies on a constant cross-section.
A pyramid narrows towards its vertex.
A pyramid’s parallel cross-sections change size, so one fixed end area cannot be used throughout.
A solid has a circular base and tapers to one point. Is it a prism or a cylinder? Explain.
2 marks
Its cross-sections parallel to the base become smaller.
A prism or cylinder has a constant cross-section.
Neither; it is a cone.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Circumference and area of circles; perimeter/area of composite shapes; choosing and converting area units; cross-sections of prisms; prism volume and capacity; mixed measurement applications.

Official curriculum source ↗

Cross-sections of prisms

In this lesson

Understand the idea.

A shape carried through space

Why cross-sections matter

See the method in action.

Choose the right shape

Identify the repeated ends.

Choose a cut parallel to an end.

State the shape and size of the section.

Calculate a cross-sectional area

Identify the triangle's two required dimensions.

Apply the triangle area formula.

Calculate and label the area.

Try it yourself.

Check your understanding

Keep practising