Cross-Sections of 3D Figures Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cross-Sections of 3D Figures.

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 formed when a plane intersects (slices through) a three-dimensional figure.

Imagine slicing a loaf of bread—each slice reveals a 2D shape. The shape you see depends on the angle and position of your cut. Slice a cylinder straight across and you get a circle; slice it at an angle and you get an ellipse. Slice a rectangular prism and you can get rectangles, triangles, or even hexagons depending on the cut.

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: The cross-section depends on both the 3D figure and the angle/position of the cutting plane. The same solid can produce very different cross-sections.

Common stuck point: Visualizing 3D intersections is hard from a 2D drawing. Use physical models (clay, Play-Doh) or interactive 3D software to build intuition.

Worked Examples

Example 1

easy
A horizontal plane cuts through the middle of a right circular cone (parallel to the base). What 2D shape is the cross-section, and how does its size compare to the base?

Solution

  1. 1
    Step 1: Visualize a right circular cone: a circular base tapering to a point (apex) at the top.
  2. 2
    Step 2: A horizontal cut parallel to the base produces a cross-section that is a circle (since every horizontal level of a cone is circular by symmetry).
  3. 3
    Step 3: A cut at the midpoint of the height means the cut is at half the total height. By similar triangles, the radius of the cross-section equals half the base radius.
  4. 4
    Step 4: Therefore the cross-section is a circle with radius = \frac{r}{2}, and its area = \pi\left(\frac{r}{2}\right)^2 = \frac{\pi r^2}{4}, which is one-quarter of the base area.

Answer

The cross-section is a circle with radius \frac{r}{2} and area \frac{1}{4} of the base area.
Horizontal cross-sections of a cone are always circles. By similar triangles, a cut at half the height gives a circle with half the base radius. Since area scales with the square of the radius, the cross-sectional area is (1/2)² = 1/4 of the base area.

Example 2

medium
Identify and describe the cross-sections formed when a plane cuts a cube in the following ways: (a) parallel to a face, (b) diagonally through four edges (cutting midpoints of four parallel edges), (c) through three vertices not on the same face.

Example 3

hard
A regular hexagonal prism is cut by a plane perpendicular to its bases that passes through two opposite edges. What is the shape of the cross-section?

Practice Problems

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

Example 1

easy
A plane cuts through a sphere. What shape is the cross-section, and when is it the largest possible cross-section?

Example 2

hard
A rectangular prism (box) has dimensions 4 cm \times 6 cm \times 8 cm. A plane cuts through the prism diagonally, connecting midpoints of the four longest edges (the edges of length 8 cm). Describe and find the area of the resulting cross-section.
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Background Knowledge

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

shapesvolumetrianglescircles