Volume of Pyramids Cones and Spheres Worked Examples Cheat Sheet

Key Facts

• The volume of a pyramid is $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the perpendicular height.
• For a rectangular pyramid, the base area is $B = lw$, so the volume is $V = \frac{1}{3}lwh$.
• The volume of a cone is $V = \frac{1}{3}\pi r^2h$, where $r$ is the radius of the circular base and $h$ is the perpendicular height.
• The volume of a sphere is $V = \frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.
• If the diameter is given, find the radius first using $r = \frac{d}{2}$.
• Pyramids and cones have exactly one third the volume of a prism or cylinder with the same base area and height.
• Volume is measured in cubic units, such as $\text{cm}^3$, $\text{m}^3$, or $\text{in}^3$.
• When using $\pi$, keep answers in exact form such as $36\pi\,\text{cm}^3$ unless the problem asks for a decimal approximation.

Vocabulary

Volume

Volume is the amount of three-dimensional space inside a solid, measured in cubic units.

Pyramid

A pyramid is a solid with one polygon base and triangular faces that meet at one vertex.

Cone

A cone is a solid with one circular base and a curved surface that narrows to one vertex.

Sphere

A sphere is a perfectly round three-dimensional shape whose points are all the same distance from the center.

Radius

The radius is the distance from the center of a circle or sphere to its edge.

Base Area

Base area is the area of the face or region used as the base of a three-dimensional figure.