Cross-Sections of 3D Solids
A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.
A cross-section is the 2D shape that appears when you slice a 3D solid with a flat plane. The slicing angle changes the shape. A horizontal slice through a cylinder is a circle; an angled slice can be an ellipse. A vertical slice through a sphere is always a circle (sometimes a great circle if it passes through the center). Slicing a cube perpendicular to a face gives a square; on a diagonal it can give a rectangle, triangle, or even a hexagon.
What you'll learn
- Predict the shape of a planar cross-section of a common 3D solid
- Recognize that the slicing angle and direction control the cross-section shape
- Identify special cross-sections (great circle of a sphere, diagonal of a cube)
Worked example
Problem. What is the cross-section of a sphere cut by ANY plane that passes through its center?
- Any plane through the center of a sphere intersects it in the largest possible circle.
- This circle has the same radius as the sphere — called a great circle.
- Connect the result back to Cross-Sections of 3D Solids so the geometric relationship is explicit.
- Check the result against the original diagram, units, or proof statement.
Answer: A great circle
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. A horizontal slice through a cylinder (perpendicular to its axis) gives:
Choices: Circle · Rectangle · Triangle
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- A perpendicular slice through a cylinder produces its base shape — a circle.
Answer: Circle
2. A vertical slice through a cylinder (parallel to its axis) gives:
Choices: Circle · Rectangle · Triangle
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- A parallel-to-axis slice gives the cylinder's profile — a rectangle.
Answer: Rectangle
3. A horizontal slice through a cone (perpendicular to its axis) gives:
Choices: Circle · Triangle · Trapezoid
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- Each level of a cone is a circle (smaller as you move toward the apex).
Answer: Circle
4. A vertical slice through the apex of a cone gives:
Choices: Circle · Triangle · Trapezoid
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- The slice exposes the cone's side profile through its apex — an isosceles triangle.
Answer: Triangle
5. Any planar slice of a sphere is:
Choices: A circle · An ellipse · A square
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- All planar slices of a sphere are circles (some larger, some smaller).
Answer: A circle
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