Every volume formula below is either base area times height, or one third of that. Knowing which family a solid belongs to is most of the work.
The formulas
| Solid | Formula | Family |
|---|---|---|
| Rectangular prism | length x width x height | base area times height |
| Any prism | base area x height | base area times height |
| Cylinder | pi x radius squared x height | base area times height |
| Pyramid | one third x base area x height | one third of the matching prism |
| Cone | one third x pi x radius squared x height | one third of the matching cylinder |
| Sphere | four thirds x pi x radius cubed | its own case |
Where the one third comes from
Three pyramids with the same base and height as a given prism fill that prism exactly. The classic classroom demonstration pours rice or water from a cone into the cylinder with the same base and height, and it takes three cone loads to fill it. The factor of one third is a measured fact about space, not an arbitrary constant.
Worked examples
Cylinder, radius 3, height 10
base area = pi x 3 x 3 = 9 pi volume = 9 pi x 10 = 90 pi about 282.7 cubic units
Cone, radius 3, height 10
one third of the cylinder above volume = 30 pi about 94.2 cubic units
Square pyramid, base 6 by 6, height 9
base area = 36 volume = 36 x 9 / 3 = 108 cubic units
Sphere, radius 3
volume = 4/3 x pi x 27 = 36 pi about 113.1 cubic units
Common mistakes
Using the slant height of a cone or pyramid
Volume uses the perpendicular height from the base to the apex. Slant height belongs to surface area problems, and mixing them up is the most expensive error in this topic.
Squaring the diameter
Halve to the radius first. Using the diameter makes a cylinder four times too large.
Forgetting the one third
A cone answer that matches the cylinder answer is missing its factor of one third.
Check yourself
Cylinder, radius 5, height 4.
100 pi, about 314.2 cubic units.
Cone with the same radius and height as that cylinder.
One third of it: 100 pi / 3, about 104.7 cubic units.
Triangular prism with a base area of 24 and a length of 11.
264 cubic units. Base area times height applies to any prism.
Sphere of radius 6.
4/3 x pi x 216 = 288 pi, about 904.8 cubic units.
See also
Sources
- Standard volume formulas as taught in United States middle grades geometry. Numerical values are rounded to one decimal place. Examples on this page are our own.