Skill: 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.