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 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.Cylinder cut by a horizontal plane halfway up its height
Answer
The cross-section is a circle; area =25π≈78.5 cm2.
First step
1
Step 1: A horizontal plane cuts perpendicular to the axis of the cylinder.
Full solution
2
Step 2: The cross-section is a circle with the same radius as the cylinder: r=5 cm.
3
Step 3: Area =πr2=π(5)2=25π≈78.5 cm2.
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 ×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?
Example 4
medium
A cube of side 6 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 8 cm and height 10 cm. It is sliced parallel to the base at height 4 cm from the base. Find the side length of the cross-section.
Example 6
medium
A sphere of radius 13 cm is sliced by a plane 5 cm from its center. Find the radius of the cross-section.Sphere of radius 13 cm; find the cross-section radius when the cut is 5 cm from the center
Example 7
medium
A triangular prism has equilateral triangle base of side 6 cm and length 10 cm. What is the area of the cross-section parallel to the base?
Example 8
hard
A sphere of radius 10 is sliced into two pieces of equal volume by a plane. How far is the plane from the center?Sphere of radius 10 sliced into two equal volumes — 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 10 and height 12. Find the area of a horizontal cross-section halfway up.
Example 11
hard
A sphere of radius r is cut by two parallel planes equidistant from the center, each at distance d from it. Find the ratio of the cross-section areas.
Example 12
challenge
A cube of side 1 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 6 and height 9. By Cavalieri's principle, what is the area of a horizontal cross-section at height h above the base, expressed as a function of h?Cone with base radius 6 and height 9; express cross-section area as a function of height h
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?Sphere of radius 7 cm sliced through its center
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).Cone with base radius 6 cm and height 12 cm; horizontal cut at height 8 cm
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?Sphere of radius 5 sliced 3 units from its center — find the cross-section radius
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?Cube of side 6 sliced through three vertices on edges meeting at one corner
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?Sphere of radius 13; cross-section circle has radius 12 — find the distance from the center
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 4 cm is sliced parallel to one face. What is the area of the cross-section?Cube of side 4 cm; horizontal cross-section parallel to one face
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 4 cm is cut by a plane through its axis (vertical slice through the center). If the height is 10 cm, find the area of the cross-section.Cylinder with radius 4 cm and height 10 cm; vertical axial cross-section
Example 28
medium
A cone has base radius 9 and height 15. It is sliced parallel to the base at height 5 from the base. Find the cross-section radius.Cone with base radius 9 and height 15; horizontal cut at height 5 from base
Example 29
medium
A rectangular prism is 3×5×7 cm. What is the largest possible area of a planar cross-section parallel to one of its faces?Rectangular prism 3 × 5 × 7 cm — which parallel face cut gives the largest cross-section?
Example 30
medium
A cube of side 4 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 12 and height 20. A horizontal cross-section has area 9π. How far above the base is this cross-section?Cone with base radius 12 and height 20; horizontal cross-section area
Example 32
hard
A cylinder of radius 5 and height 20 is cut by a plane tilted at 45° to its axis passing through the center. What shape is the cross-section?
Example 33
hard
A cone has a base radius of 6 cm. A plane perpendicular to the base passes through the center, cutting the cone into two equal pieces. If the cone's height is 9 cm, find the area of this cross-section.Cone with radius 6 cm and height 9 cm; vertical cross-section through the axis