Lesson 70: Volume

New Concept

Volume describes the space an object occupies, measured in cubic units. The formula for the volume ($V$) of a rectangular prism is:

What’s next

To start, we'll explore volume more deeply. Next, you’ll apply the volume formula in worked examples and practice problems involving rectangular prisms and other solids.

Property

Volume is the space occupied by a shape. We measure it using cubic units, such as $1 \text{ cm}^3$ (one cubic centimeter), $1 \text{ ft}^3$ (one cubic foot), or $1 \text{ m}^3$ (one cubic meter).

Examples

A rectangular prism constructed with 4 layers of cubes, where each layer has 3 rows of 5 cubes, has a volume of $4 \times (3 \times 5) = 60$ cubes.
A box that holds 25 cubes on its base and is 4 layers high has a volume of $25 \times 4 = 100$ cubes.

Explanation

Think of volume as counting how many tiny building blocks, or unit cubes, fit inside a 3D shape. First, you figure out how many cubes cover the bottom layer, then you multiply that by the number of layers. It's like stacking pancakes, but with cubes! A shape's volume is its total cube count.

Property

The formula to find the volume ($V$) of a rectangular prism with length ($l$), width ($w$), and height ($h$) is:

Examples

A box with dimensions 6 cm long, 4 cm wide, and 2 cm high has a volume of $V = 6 \text{ cm} \times 4 \text{ cm} \times 2 \text{ cm} = 48 \text{ cm}^3$.
A prism with dimensions 10 feet, 5 feet, and 4 feet has a volume of $V = 10 \text{ ft} \times 5 \text{ ft} \times 4 \text{ ft} = 200 \text{ ft}^3$.

Explanation

Forget counting every single cube! Just multiply the three key measurements: length, width, and height. This formula is a super-fast shortcut to find the total number of cubes that would fill up any box-shaped object. It's the ultimate secret code for finding the space inside a rectangular prism. It is truly math magic!