Lesson 6 — Cylinders and Cones
Big idea: Cylinders and cones are the circular counterparts of prisms and pyramids, so their volume formulas have the same shape with B = πr².
Words to know
| Word | Meaning |
|---|---|
| Cone | The solid formed by all segments connecting a point P outside a plane to the points of a circular region R in that plane. |
| Circular region | The union of a circle and its interior. |
| Base | The circular face of a cone or cylinder. |
| Apex | The point where all the segments of a cone meet. |
| Lateral surface | The curved surface of a cone or cylinder. |
| Altitude | The perpendicular distance between the planes of the bases. |
| Axis | For a cone, the segment from the apex to the center of the base; for a cylinder, the segment joining the centers of the bases. |
| Right / oblique cone or cylinder | Right if the axis is perpendicular to the base; oblique otherwise. |
| Cylinder | The solid formed by all segments parallel to a line l that connect a circular region in one plane to a second parallel plane. |
Concepts to know
- A cone is like a pyramid with a circular base and one curved lateral surface instead of flat triangular faces.
- A cylinder is like a prism with two circular bases and one curved lateral surface.
- The altitude of a cylinder is the same idea as the altitude of a prism.
- Cylinders and cones are classified as right or oblique by the direction of their axes.
- A cylinder can be approximated by a prism, and a cone by a pyramid, so the prism and pyramid volume formulas carry over.
- Theorem 81: The volume of a cylinder is V = Bh = πr²h.
- Theorem 82: The volume of a cone is V = ⅓Bh = ⅓πr²h.
Formulas / rules
- Cylinder volume: V = πr²h
- Cone volume: V = ⅓πr²h
- Base area of either: B = πr²
- Lateral area (developed in the exercises): cylinder L.A. = 2πrh; cone L.A. = πrℓ, where ℓ is the slant height.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a cylinder’s volume | Square the radius, multiply by π and the height. |
| Find a cone’s volume | Use the cylinder formula and divide by 3. |
| Given the diameter | Halve it first: r = d ÷ 2. |
| Find a missing height | Substitute V and r, then solve for h. |
| Find lateral area of a cylinder | Unroll it into a rectangle: 2πr × h. |
| Compare a cone and a cylinder | Same base and height: the cone is ⅓ the cylinder’s volume. |
Watch out
- Use the radius, not the diameter, in πr².
- The height is perpendicular to the base; a slant height is longer.
- Do not forget the ⅓ for a cone.
- πr²h is cubic units; 2πrh is square units.