Integrated Math 3 / Unit 7: Geometric Modeling
7.4 Cross-Sections and Solids of Revolution
Notes
A cross-section is the shape you see when a plane slices a solid.
- Cylinder: parallel to the base gives a circle; through the axis gives a rectangle.
- Cone: parallel to the base gives a circle; through the apex and perpendicular to the base gives a triangle.
- Sphere: every slice is a circle. At distance $d$ from the center, the circle has radius $\displaystyle \sqrt{R^2 - d^2}$.
- A solid of revolution forms when a 2D shape spins around a line. A rectangle spun about one side gives a cylinder. A right triangle spun about a leg gives a cone.
Practice
A plane cuts a cylinder through its axis, perpendicular to the bases. What is the cross-section?
The slice contains the full height and the full diameter.
A right triangle with legs 3 and 4 is spun around the leg of length 4. What is the volume of the solid?
The solid is a cone with $r = 3$ and $h = 4$: $\displaystyle \frac{1}{3}\pi(9)(4) = 12\pi$.
A sphere has radius 5. A plane cuts it 3 units from its center. What is the area of the circular cross-section?
The circle's radius is $\displaystyle \sqrt{25 - 9} = 4$, so the area is $16\pi$.
Ready for more? Take the Unit 7 test.