Geometry Cross Sections and Solids of Revolution Lesson (Free PDF)
This geometry cross sections lesson for Geometry covers cross sections and solids of revolution (Common Core G-GMD.B.4), with a free PDF to print.
A cross section is the shape made when a plane slices a solid. A cylinder cut parallel to its base gives a circle, a cut through its axis gives a rectangle, and every sphere section is a circle. Rotating a rectangle about a side makes a cylinder, and rotating a right triangle about a leg makes a cone.
Geometry · Common Core State Standards: G-GMD.B.4
Key facts in this geometry cross sections lesson
- Cylinder: parallel cut is a circle; cut through the axis is a rectangle.
- Cube: cut parallel to a face is a square.
- Cone: parallel cut is a circle; cut through the apex is a triangle.
- Sphere: every plane cross section is a circle.
- Rotating a rectangle about a side gives a cylinder; a right triangle about a leg gives a cone.
What is a cross section?
A cross section is the two-dimensional shape you see when a plane slices through a three-dimensional solid, like cutting a loaf of bread. The shape depends on the solid and on the angle of the cut.
What are the cross sections of a cylinder?
A plane parallel to the bases gives a circle. A plane perpendicular to the bases through the axis gives a rectangle. A slanted plane gives an ellipse.
What are the cross sections of a cube and a cone?
A plane parallel to a face of a cube gives a square. A plane through two opposite edges gives a rectangle, and a plane through three corners gives an equilateral triangle. For a cone, a plane parallel to the base gives a circle and a plane through the apex and axis gives a triangle.
| Solid | Cut | Cross section |
|---|---|---|
| Cylinder | parallel to base | circle |
| Cylinder | through the axis | rectangle |
| Cube | parallel to a face | square |
| Cone | parallel to base | circle |
| Cone | through the apex | triangle |
| Sphere | any plane | circle |
The cross section of a sphere is always a circle. A plane at distance \(d\) from the center of a sphere of radius \(r\) cuts a circle of radius \(\sqrt{r^{2} – d^{2}}\).
What is a solid of revolution?
When you rotate a two-dimensional shape about a line, the region it sweeps out is a solid of revolution. A rectangle rotated about one side makes a cylinder, a right triangle rotated about a leg makes a cone, and a semicircle rotated about its diameter makes a sphere.
- Sketch the shape and the axis of rotation.
- Decide which side lies on the axis: it becomes the height or the axis of the solid.
- The distance from the axis becomes the radius.
- Use the volume formula for the solid you found.
Example: A right triangle with legs 6 and 8 rotated about the leg of length 8 makes a cone with radius 6 and height 8, with volume \(\frac{1}{3}\pi \cdot 6^{2} \cdot 8 = 96\pi\).
Geometry cross sections lesson: questions and answers
What is a cross section in geometry?
A cross section is the plane figure formed where a plane cuts through a solid. For a cylinder cut parallel to its base it is a circle.
What cross sections can a cube have?
A cube can have a square, a rectangle, a triangle, a pentagon or a hexagon as a cross section, depending on how the plane cuts it.
What solid forms when a right triangle rotates about a leg?
A cone. The leg on the axis is the height and the other leg is the radius.
How do you find the radius of a circular cross section of a sphere?
If the plane is d units from the center of a sphere with radius r, the circle has radius the square root of r squared minus d squared.
What is a solid of revolution?
It is a three-dimensional shape formed by rotating a two-dimensional region about a line, such as a cylinder made from a rectangle.
How to use this geometry cross sections lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the geometry cross sections lesson as a PDF with the button below and keep it with your notes.
Download the geometry cross sections lesson as a PDF with the button below and keep it with your notes.
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