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

A cross-section is the flat, two-dimensional shape revealed when a plane cuts through a three-dimensional solid. For example, slicing a cylinder parallel to its base gives a circle, while slicing it at an angle gives an ellipse.

Imagine slicing a loaf of breadβ€”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 β†’

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: A cross-section is the two-dimensional shape made by cutting through a three-dimensional solid.

Common stuck point: The procedure for cross-sections of 3d figures is the easy part; the trap is naming the original solid. Asking "What two-dimensional shape is exposed by the slice?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: What two-dimensional shape is exposed by the slice?

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?

Answer

The cross-section is a circle with radius r2\frac{r}{2} and area 14\frac{1}{4} of the base area.

First step

1
Step 1: Visualize a right circular cone: a circular base tapering to a point (apex) at the top.

Full solution

  1. 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).
  2. 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.
  3. 4
    Step 4: Therefore the cross-section is a circle with radius =r2= \frac{r}{2}, and its area =Ο€(r2)2=Ο€r24= \pi\left(\frac{r}{2}\right)^2 = \frac{\pi r^2}{4}, which is one-quarter 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)Β² = 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?

Example 4

medium
A right circular cone has base radius 66 and height 99. A plane parallel to the base cuts the cone at a height 33 above the base. Find the area of the cross-section.

Example 5

medium
A cube has edge length 66. A plane cuts through three vertices that share a single corner vertex's neighbors (i.e., the three vertices adjacent to one vertex). Describe the cross-section and find its area.

Example 6

medium
A cone has height 1212 and base radius 88. A plane parallel to the base cuts the cone halfway up (at height 66). What is the ratio of the area of the cross-section to the area of the base?

Example 7

hard
A cube of edge length 22 is cut by a plane that passes through the midpoints of six edges, forming a regular hexagonal cross-section. What is the side length of the hexagon?

Example 8

hard
A cylinder has radius 55 and height 1010. A plane cuts the cylinder so that it enters one circular face at a point and exits the opposite circular face at a point directly above the opposite side. Describe the cross-section.

Example 9

hard
A sphere of radius 1010 has two parallel cross-sections of radii 66 and 88 on opposite sides of the center. What is the distance between the two cutting planes?

Example 10

challenge
A unit cube is cut by a plane through three midpoints of edges that all meet at one vertex (i.e., the midpoints of the three edges emanating from a single corner). What is the area 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 44 cm Γ—6\times 6 cm Γ—8\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.

Example 3

easy
A cube of edge length 44 cm is sliced by a plane parallel to one face. What is the area of the cross-section?

Example 4

easy
A sphere of radius 55 is cut by a plane through its center. What is the area of the cross-section?

Example 5

medium
A cylinder has radius 33 and height 1010. What is the area of a vertical cross-section that passes through the cylinder's central axis?

Example 6

medium
A rectangular prism has dimensions 3Γ—4Γ—53 \times 4 \times 5. A plane cuts the prism parallel to the 3Γ—43 \times 4 face. What is the area of the cross-section?

Example 7

medium
A sphere has radius 1313. A horizontal plane cuts the sphere 55 units above the center. What is the radius of the cross-section?

Example 8

hard
A right circular cone has base radius 66 and height 88. A plane through the apex cuts the base along a chord of length 1010. What is the area of the resulting triangular cross-section?

Example 9

hard
A cube of side length aa is cut by a plane through three adjacent vertices around a single corner. What is the perimeter of the resulting triangle?

Example 10

hard
A cone has base radius 99 and height 1212. A horizontal cross-section has area 4Ο€4\pi. How far above the base is the cutting plane?
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Background Knowledge

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

shapesvolumetrianglescircles