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All courses Geometry · G-GMD.3 23 of 57
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Find the volume of a cylinder

Problem
A cylinder has \(\text{radius}~3\) and \(\text{height}~10\). Find its volume.
Unworked solid diagram showing only the supplied dimensions and unknowns. Open full size
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Hint

Use the cylinder volume formula \(V=\pi r^2h\).

Square the radius first, then multiply by the height and by \(\pi\).

Solution walkthrough

01

Choose the cylinder relationship

\[V=\pi~r^2~h\]

A cylinder has congruent circular cross-sections through its perpendicular height, so volume equals circular base area pi r squared times height h.

02

Source the two dimensions

\[r=3;~h=10\]

The prompt and inspected diagram label the base radius as 3 and the perpendicular height as 10. The radius, not the diameter, belongs in the formula.

03

Substitute after naming the formula

\[V=\pi(3)^2(10)=\pi(9)(10)\]

The radius is squared first, giving 9. The height factor remains 10.

04

Calculate and check dimensions

\[V=90\pi~\text{cubic}~\text{units};~\text{units}^2~\text{times}~\text{units}=\text{units}^3\]

Multiplying 9 by 10 gives 90. Since base area has square units and height has linear units, the result correctly has cubic units.

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Another way

  1. Find the base area first, pi(3 squared)=9pi square units, then multiply by height 10 to obtain 90pi cubic units.

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Common mistake

Do not use V=pi r h. The radius must be squared because pi r squared is the area of the circular base.

Completed solid diagram with formula, substitution, and exact volume conclusion.
90π

Solution walkthrough video