Cross-Sections and Solids of Revolution
Slicing a solid reveals 2D shapes; spinning a shape builds a solid.
About this topic
A cross-section is the shape you see when a plane slices through a solid: slicing a cylinder gives a circle or rectangle, and slicing a cone can give a circle, ellipse, parabola or hyperbola. A solid of revolution is made by spinning a 2D shape around an axis, so a rectangle gives a cylinder and a semicircle gives a sphere.
Thinking in slices is the key to calculating volumes in calculus, and to understanding medical scans such as CT and MRI.
Worked example: slice a cube parallel to a face and the cross-section is a square; slice it at a tilt and you can get a rectangle, triangle or even a hexagon. Spin a right triangle with legs 3 and 4 about the leg of length 4 and you get a cone with radius 3 and height 4. Spinning a rectangle around its side creates a cylinder, like a potter's wheel turning clay.
Doctors read CT slices to build a 3D view, and engineers use cross-sections in blueprints. Students sometimes think a cross-section must be the same shape as the base, but angled cuts change it. Visualise by imagining a knife passing through a loaf of bread, a block of cheese or a fruit. For revolution, ask what shape is traced by each point, usually a circle, and pay attention to which line is the axis.
Key ideas
- Slice a cube to get squares, rectangles or hexagons
- Rotating a rectangle gives a cylinder
- Rotating a triangle gives a cone
Quick check
Use cases: where Cross-Sections and Solids of Revolution helps
- Homework: review the key idea behind Cross-Sections and Solids of Revolution before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Cross-Sections and Solids of Revolution without notes.
- Visual learning: play with the interactive lab to see how Cross-Sections and Solids of Revolution changes when you adjust its inputs.
- Teaching a friend: explain Cross-Sections and Solids of Revolution in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Cross-Sections and Solids of Revolution?
Slicing a solid reveals 2D shapes; spinning a shape builds a solid.
What will I learn in Cross-Sections and Solids of Revolution?
Slice a cube to get squares, rectangles or hexagons. Rotating a rectangle gives a cylinder. Rotating a triangle gives a cone.
Is Cross-Sections and Solids of Revolution part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Cross-Sections and Solids of Revolution lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Cross-Sections and Solids of Revolution?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.