Area & Volume

Volume of Solids

Finding the volume of prisms, cylinders, pyramids, cones, and spheres.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Surface Area of Solids

Why a Cone Holds Exactly a Third

A cone and a cylinder share the same circular base and the same height. Before reading on, predict: if you filled the cone with sand and poured it into the cylinder, how many cone-fuls do you think it would take to fill the cylinder completely?

It takes exactly three cone-fuls, every time, regardless of the base size or height — a genuinely surprising fact that can be verified experimentally, and is provable rigorously using calculus in a later course. For now, treat it as an established result: a cone's volume is always exactly one-third of the cylinder sharing its base and height, and a pyramid's volume is likewise exactly one-third of the matching prism.

Definition — Volume Formulas for Prisms, Cylinders, Pyramids, and Cones

A prism or cylinder's volume equals its base area times its height: V = Bh. A pyramid or cone's volume is exactly one-third of the matching prism or cylinder: V = (1/3)Bh.

This one-third relationship connects to a deeper idea called Cavalieri's Principle: if two solids have the same height, and at every single height their cross-sectional slice has the same area, the two solids must have exactly the same volume — even if the solids look completely different in shape. This principle is what allows volume formulas derived for one shape (like a right cylinder) to extend to oblique or irregular versions with the same base and height.

Worked Example — Finding a Prism's Volume

A rectangular prism has a base area of 24 square units and a height of 9. Find its volume. V = Bh = 24 × 9 = 216 cubic units.

Worked Example — Finding a Cone's Volume

A cone has a base radius of 6 and a height of 10. Find its volume. First find the base area: B = π(6²) = 36π. Apply the cone formula: V = (1/3)(36π)(10) = 120π ≈ 376.99 cubic units.

A sphere's volume, (4/3)πr³, follows its own distinct formula rather than a base-times-height pattern, since a sphere has no flat base at all. One historically famous fact, attributed to Archimedes, connects it back to familiar shapes anyway: a sphere's volume is always exactly two-thirds of the volume of the smallest cylinder that tightly contains it.

Worked Example — Finding a Sphere's Volume

A sphere has radius 5. Find its volume. V = (4/3)π(5³) = (4/3)π(125) = (500/3)π ≈ 523.6 cubic units.

V_{\text{cone/pyramid}} = \dfrac{1}{3}Bh \qquad V_{\text{sphere}} = \dfrac{4}{3}\pi r^{3}

Tip

Before finalizing a cone or pyramid volume, sanity check it against the matching prism or cylinder's volume — the cone or pyramid's answer should always come out to roughly a third of that larger value, a quick way to catch a forgotten factor of 1/3.

Common Mistakes

Forgetting to multiply by 1/3 when finding a cone or pyramid's volume, using the plain prism/cylinder formula instead.
Cones and pyramids always take exactly one-third of the volume their matching prism or cylinder would have — that factor is never optional.
Using a sphere's diameter in place of its radius in the volume formula.
The sphere volume formula, (4/3)πr³, specifically uses the radius — if only the diameter is given, divide it by 2 first before cubing.

Key Takeaways

A prism or cylinder's volume is base area times height; a pyramid or cone's volume is exactly one-third of that.
Cavalieri's Principle explains why matching cross-sectional areas at every height guarantee equal volumes, even for differently shaped solids.
A sphere's volume, (4/3)πr³, is exactly two-thirds of its tightest-fitting cylinder's volume.

Summary

Volume formulas for pyramids, cones, and spheres connect back to prisms and cylinders through consistent, provable ratios. The final unit shifts from measuring solids to proving geometric facts using coordinates and algebra.

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