Cross-Section Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cross-Section.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The two-dimensional shape that is revealed when a three-dimensional solid is sliced through by a flat plane.

Slice an orange—the cut surface is a cross-section (a circle).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Cross-sections reveal internal structure; shape depends on angle of cut.

Common stuck point: Different cuts of the same object give different cross-sections.

Sense of Study hint: Try slicing clay or Play-Doh at different angles and examine each flat cut surface to see the different shapes.

Worked Examples

Example 1

easy
A cylinder of radius 5 cm and height 12 cm is cut by a horizontal plane halfway up its height. Describe and find the area of the cross-section.

Solution

  1. 1
    Step 1: A horizontal plane cuts perpendicular to the axis of the cylinder.
  2. 2
    Step 2: The cross-section is a circle with the same radius as the cylinder: r = 5 cm.
  3. 3
    Step 3: Area = \pi r^2 = \pi (5)^2 = 25\pi \approx 78.5 cm^2.

Answer

The cross-section is a circle; area = 25\pi \approx 78.5 cm^2.
Any plane perpendicular to the axis of a cylinder produces a circular cross-section identical to the base. The height at which the cut occurs does not affect the shape or size, since the cylinder has constant cross-section along its axis.

Example 2

medium
A square pyramid with a 6 cm \times 6 cm base and height 9 cm is cut by a horizontal plane 3 cm above the base. Find the shape and dimensions of the cross-section.

Example 3

medium
A cone is cut by a plane parallel to its base. What shape is the cross-section?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What shape is the cross-section when a sphere is cut by any plane through its centre? What is the area if the sphere has radius 7 cm?

Example 2

hard
A cone with base radius 6 cm and height 12 cm is cut by a plane parallel to the base at height 8 cm from the base. Find the radius and area of the cross-section, then compute the ratio of areas (cross-section : base).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

planeshapes