Calculate the volume of pyramids and cones

Volume of a pyramid or cone is the amount of three-dimensional space enclosed, calculated as one-third of the product of its base area and perpendicular height: $V=\frac{1}{3}Bh$ , including $V=\frac{1}{3}\pi r^2h$ for a circular cone. Learners distinguish vertical height from slant height, select appropriate units, and interpret cubic units; the scope is limited to standard measurable pyramids and right circular cones, excluding calculus-based or more generalized treatments.

Detailed Explanation: Calculate the volume of pyramids and cones

To find the volume of a pyramid or cone, use

V=\frac{1}{3}Bh

where:

$V$ is the volume,
$B$ is the area of the base,
$h$ is the perpendicular (vertical) height.

The slant height is not used in the volume formula.

For a cone, the base is a circle, so $B=\pi r^2$ . Therefore,

V=\frac{1}{3}\pi r^2h

Example: Square pyramid

A square pyramid has a base side length of $6\text{ cm}$ and a vertical height of $8\text{ cm}$ . Find its volume.

Step 1: Find the base area.

The base is a square:

B=6\times 6=36\text{ cm}^2

Step 2: Identify the perpendicular height.

The vertical height is $8\text{ cm}$ . Use this height, not the slant height.

Step 3: Substitute into the formula.

V=\frac{1}{3}Bh

V=\frac{1}{3}(36)(8)

Step 4: Calculate.

V=96\text{ cm}^3

The volume of the pyramid is

\boxed{96\text{ cm}^3}

Volume is measured in cubic units because it measures three-dimensional space.

Learn by doing: Calculate the volume of pyramids and cones

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