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 orangeβ€”the cut surface is a cross-section (a circle).

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 2D shape you see on the cut when a plane slices through a solid.

Common stuck point: The procedure for cross-section is the easy part; the trap is assuming the cut shape equals the face shape. Asking "Am I finding the flat 2D shape exposed when a plane cuts a 3D solid?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I finding the flat 2D shape exposed when a plane cuts a 3D solid?

Worked Examples

Example 1

easy
A cylinder of radius 55 cm and height 1212 cm is cut by a horizontal plane halfway up its height. Describe and find the area of the cross-section.

Answer

The cross-section is a circle; area =25Ο€β‰ˆ78.5= 25\pi \approx 78.5 cm2^2.

First step

1
Step 1: A horizontal plane cuts perpendicular to the axis of the cylinder.

Full solution

  1. 2
    Step 2: The cross-section is a circle with the same radius as the cylinder: r=5r = 5 cm.
  2. 3
    Step 3: Area =Ο€r2=Ο€(5)2=25Ο€β‰ˆ78.5= \pi r^2 = \pi (5)^2 = 25\pi \approx 78.5 cm2^2.
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 66 cm Γ—\times 66 cm base and height 99 cm is cut by a horizontal plane 33 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?

Example 4

medium
A cube of side 66 is sliced by a plane perpendicular to a face diagonal passing through the cube's center. Describe the cross-section shape.

Example 5

medium
A square pyramid has base side 88 cm and height 1010 cm. It is sliced parallel to the base at height 44 cm from the base. Find the side length of the cross-section.

Example 6

medium
A sphere of radius 1313 cm is sliced by a plane 55 cm from its center. Find the radius of the cross-section.

Example 7

medium
A triangular prism has equilateral triangle base of side 66 cm and length 1010 cm. What is the area of the cross-section parallel to the base?

Example 8

hard
A sphere of radius 1010 is sliced into two pieces of equal volume by a plane. How far is the plane from the center?

Example 9

hard
A unit cube is sliced by a plane through one edge and a non-adjacent vertex on the opposite face. Describe the cross-section.

Example 10

hard
A square pyramid has base side 1010 and height 1212. Find the area of a horizontal cross-section halfway up.

Example 11

hard
A sphere of radius rr is cut by two parallel planes equidistant from the center, each at distance dd from it. Find the ratio of the cross-section areas.

Example 12

challenge
A cube of side 11 is sliced by a plane through the midpoints of six edges, forming a regular hexagonal cross-section. Find the area of that hexagon.

Example 13

challenge
A cone has base radius 66 and height 99. By Cavalieri's principle, what is the area of a horizontal cross-section at height hh above the base, expressed as a function of hh?

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 77 cm?

Example 2

hard
A cone with base radius 66 cm and height 1212 cm is cut by a plane parallel to the base at height 88 cm from the base. Find the radius and area of the cross-section, then compute the ratio of areas (cross-section : base).

Example 3

easy
You slice an orange straight across the middle. What shape is the cut surface?

Example 4

easy
A cube is sliced parallel to one of its faces. What is the cross-section?

Example 5

easy
A cylinder is sliced parallel to its circular base. What is the cross-section?

Example 6

easy
A cylinder is sliced straight down through its axis. What is the cross-section?

Example 7

easy
True or false: every cross-section of a given solid is the same shape.

Example 8

easy
A cone is sliced parallel to its circular base. What is the cross-section?

Example 9

easy
A cone is sliced straight down through its apex. What shape appears?

Example 10

easy
A rectangular box is sliced parallel to one face. What is the cross-section?

Example 11

medium
A cone is sliced at a slight angle to its base (not through the apex). What shape is the cross-section?

Example 12

medium
A square-based pyramid is sliced parallel to its base. What is the cross-section?

Example 13

medium
A sphere of radius 5 is sliced 3 units from its center. What is the radius of the circular cross-section?

Example 14

medium
Why does slicing a cone give such different shapes (circle, ellipse, parabola)?

Example 15

medium
A rectangular prism (box) is sliced by a plane cutting diagonally across, corner to corner of one face and angled. What general type of polygon can result?

Example 16

medium
Architects use cross-sections of buildings. What does a cross-section drawing show that a front view does not?

Example 17

medium
A triangular prism is sliced parallel to its triangular ends. What is the cross-section?

Example 18

medium
Stacking many thin identical circular cross-sections of the same radius builds what solid?

Example 19

challenge
A cube of side 6 is sliced by a plane through three vertices that each sit on edges meeting at one corner. What shape is the cross-section?

Example 20

challenge
Explain why a plane can slice a cube into a regular hexagon, naming the maximum number of sides a cube cross-section can have.

Example 21

challenge
A sphere of radius 13 is sliced, and the circular cross-section has radius 12. How far from the center was the slice made?

Example 22

challenge
A solid is formed so that every horizontal cross-section is a square, but the squares shrink linearly to a point at the top. What solid is this, and how does its cross-section area change with height?

Example 23

easy
What shape is the cross-section when a rectangular prism is sliced parallel to its base?

Example 24

easy
A cube of side 44 cm is sliced parallel to one face. What is the area of the cross-section?

Example 25

easy
A triangular prism is sliced parallel to its triangular base. What is the cross-section?

Example 26

easy
What is the cross-section of a sphere sliced by a plane tangent to it?

Example 27

medium
A cylinder with radius 44 cm is cut by a plane through its axis (vertical slice through the center). If the height is 1010 cm, find the area of the cross-section.

Example 28

medium
A cone has base radius 99 and height 1515. It is sliced parallel to the base at height 55 from the base. Find the cross-section radius.

Example 29

medium
A rectangular prism is 3Γ—5Γ—73 \times 5 \times 7 cm. What is the largest possible area of a planar cross-section parallel to one of its faces?

Example 30

medium
A cube of side 44 is sliced by a plane through three edges, cutting each at the midpoint. What is the cross-section shape?

Example 31

hard
A cone has base radius 1212 and height 2020. A horizontal cross-section has area 9Ο€9\pi. How far above the base is this cross-section?

Example 32

hard
A cylinder of radius 55 and height 2020 is cut by a plane tilted at 45Β°45Β° to its axis passing through the center. What shape is the cross-section?

Example 33

hard
A cone has a base radius of 66 cm. A plane perpendicular to the base passes through the center, cutting the cone into two equal pieces. If the cone's height is 99 cm, find the area of this cross-section.
← Back to Cross-Section Practice Mode

Background Knowledge

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

planeshapes