Lesson 7 — Spheres
Big idea: A sphere is the set of all points in space at a given distance from a center, and both its volume and surface area depend only on its radius.
Words to know
| Word | Meaning |
|---|---|
| Sphere | The set of all points in space that are at a given distance from a given point. |
| Center | The given point from which every point of the sphere is equally distant. |
| Radius | The distance from the center to any point of the sphere; also a segment of that length. |
| Diameter | A segment through the center with both endpoints on the sphere; its length is 2r. |
Concepts to know
- The definition of a sphere is like the definition of a circle, with “in space” replacing “in a plane.”
- Center, radius, and diameter have the same meanings for spheres as for circles.
- By the volume of a sphere, we mean the volume of the solid consisting of the sphere and its interior.
- The volume formula is derived with Cavalieri’s Principle: a sphere of radius r has the same volume as a cylinder of radius r and height 2r with two cones removed.
- A cross section of the sphere at distance d from the center has area π(r² − d²).
- Theorem 83: The volume of a sphere is ⁴⁄₃π times the cube of its radius: V = ⁴⁄₃πr³.
- Theorem 84: The surface area of a sphere is 4π times the square of its radius: A = 4πr².
- The surface-area formula can be seen by dividing the sphere into many small pyramids with altitude r.
- A great circle of a sphere has area πr², so the sphere’s surface area is four times the area of a great circle (Archimedes).
Formulas / rules
- Volume: V = ⁴⁄₃πr³
- Surface area: A = 4πr²
- Diameter: d = 2r, so r = d ÷ 2
- Great-circle area: πr²
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a sphere’s volume | Cube the radius, multiply by π, then multiply by ⁴⁄₃. |
| Find a sphere’s surface area | Square the radius and multiply by 4π. |
| Given the diameter | Halve it before using either formula. |
| Find the radius from the volume | Solve V = ⁴⁄₃πr³ for r³, then take the cube root. |
| Compare surface area to a great circle | The sphere’s area is 4 times the great circle’s area. |
| Work a real-world weight problem | Find the volume first, then multiply by the density. |
Watch out
- Do not use the diameter in place of the radius.
- Volume uses r³; surface area uses r².
- The ⁴⁄₃ applies to volume only, not to surface area.
- Surface area is in square units; volume is in cubic units.
- A sphere has no flat faces, edges, or vertices.