Volume
Overview
Volume is the amount of space contained within a solid, measured in cubic units. Nearly every volume formula in this lesson has the same skeleton: find the area of the base, then multiply by the height — with one adjustment for the pointed solids, pyramids and cones, which take exactly one third of their straight-sided counterparts.
Vocabulary
- Volume
- The amount of space contained within a solid. Volume is measured in units cubed (units³).
- Prism
- A polyhedron with two congruent faces, called bases, that lie in parallel planes. Its volume is the area of a base times the height: V = Bh.
- Lateral Faces
- The faces of a prism other than the bases.
- Cylinder
- A solid with two congruent circular bases in parallel planes; treated like a prism whose base is a circle: V = πr²h.
- Pyramid and Cone
- Pointed solids whose volume is one third of the corresponding prism or cylinder: V = ⅓Bh, and for a cone, V = ⅓πr²h.
- Sphere
- The set of all points in space a fixed distance (the radius) from a center point. Its volume is V = (4/3)πr³.
Main Ideas
- A room 10 ft by 12 ft with 8-ft ceilings: V = 10 × 12 × 8 = 960 cubic feet.
- For any prism, first compute the base area B — for a triangular prism, B = ½bh of the triangular base — then multiply by the prism's length.
- Pyramids and cones take the factor ⅓. Forgetting it triples the answer.
- Working backwards works too: given the volume and the base area, the height is V ÷ B.
- Most formulas are provided on the standard formula reference sheet — the skill is choosing the right one and substituting carefully.
Lesson Notes
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Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.