Volume of Prisms and Cylinders
Develop the volume formula V = Bh where B is the base area. Apply to prisms and cylinders and solve multi-step problems involving volumes of composite 3D figures.
A lesson of Pre-Algebra, in Geometry Foundations, about 10 minutes.
Volume
By the end of this lesson, you will be able to find the volume of prisms, cylinders, pyramids, and cones.
What Is Volume?
Volume: the amount of space inside a 3D figure. Measured in cubic units.
Prisms and Cylinders
$$V = B \times h$$
where $B$ = area of the base and $h$ = height.
Rectangular prism: $B = lw \to V = lwh$
Cylinder: $B = \pi r^2 \to V = \pi r^2 h$
Example (cylinder): $r = 4$ cm, $h = 9$ cm
$V = \pi(16)(9) = 144\pi \approx 452.4$ cm³
Pyramids and Cones
$$V = \frac{1}{3}Bh$$
Pyramids and cones hold exactly $\frac{1}{3}$ as much as their corresponding prism/cylinder.
Square pyramid: $B = s^2 \to V = \frac{1}{3}s^2 h$
Cone: $B = \pi r^2 \to V = \frac{1}{3}\pi r^2 h$
Example (cone): $r = 3$ m, $h = 7$ m
$V = \frac{1}{3}\pi(9)(7) = 21\pi \approx 65.97$ m³
Sphere
$$V = \frac{4}{3}\pi r^3$$
Example: $r = 6$ units
$V = \frac{4}{3}\pi(216) = 288\pi \approx 904.8$ cubic units
Summary
Prisms/Cylinders: $V = Bh$ (base area × height).
Pyramids/Cones: $V = \frac{1}{3}Bh$ (one-third of corresponding prism/cylinder).
Sphere: $V = \frac{4}{3}\pi r^3$.