Volume (prisms, pyramids, cones, spheres, cylinders)
Volume of Solids
Volume of Solids is a topic in Three-Dimensional Measurement in the California Common Core State Standards. It is aligned to Standard G-GMD.3, which requires students to use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Volume formulas include \(Bh\) for a prism or cylinder, \(\dfrac13 Bh\) for a pyramid or cone, and \(\dfrac43\pi r^3\) for a sphere.
Theory
- Prism / cylinder: \(V=Bh\), base area times height.
- Pyramid / cone: \(V=\dfrac13 Bh\) — exactly one third of the prism or cylinder with the same base and height.
- Sphere: \(V=\dfrac43\pi r^3\).
The volume formulas:
How to find volume
- Find the base area \(B\).
- Prism/cylinder: multiply by the height.
- Pyramid/cone: multiply by the height and by \(\dfrac13\).
- Sphere: use \(\dfrac43\pi r^3\).
Volume is base area times height (length times width times height).
| \(V\) | \(=\) | \(4\times 3\times 5=60\) |
\(V=Bh=\pi r^2 h\).
| \(V\) | \(=\) | \(\pi(3)^2(7)=63\pi\) |
A cone is one third of the cylinder: \(V=\dfrac13\pi r^2 h\).
| \(V\) | \(=\) | \(\dfrac13\pi(36)(10)=120\pi\) |
Use \(V=\dfrac43\pi r^3\).
| \(V\) | \(=\) | \(\dfrac43\pi(3)^3\) |
| \(=\) | \(\dfrac43\pi(27)=36\pi\) |
Common pitfalls
Frequently asked questions
What is the volume of a prism or cylinder?
\(V=Bh\): the base area times the height.
What is the volume of a pyramid or cone?
\(V=\dfrac13 Bh\) — one third the base area times the height.
What is the volume of a sphere?
\(V=\dfrac43\pi r^3\).
Why do cones and pyramids have a factor of one third?
A cone or pyramid fills exactly one third of the cylinder or prism with the same base and height.