Practice Cross-Sections of 3D Figures in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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.

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?

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?

Example 4

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

Example 5

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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