Lesson 42: Volume

New Concept

The volume of a solid is the total amount of space occupied or enclosed by the solid. The specific formula for a rectangular prism is:

Volume of a Rectangular Prism

$$V = lwh$$

What’s next

This card is just the foundation. Next, you’ll apply the volume formula in worked examples involving rectangular prisms, cubes, and even complex composite shapes.

Property

To find the volume of any prism, multiply the area of the base ($B$) by the height ($h$).

Examples

• A triangular prism has a base area of $30 \text{ cm}^2$ and a height of $15 \text{ cm}$. $V = Bh = (30 \text{ cm}^2)(15 \text{ cm}) = 450 \text{ cm}^3$.
• An L-shaped building has a base area of $700 \text{ ft}^2$ and a height of $12 \text{ ft}$. $V = Bh = (700 \text{ ft}^2)(12 \text{ ft}) = 8400 \text{ ft}^3$.

Explanation

Think of volume as the total space inside a 3D object. The formula $V=Bh$ is a super useful shortcut. Just find the area of the bottom layer (the base), and then multiply it by how many layers tall the object is (the height). It’s like calculating the space by stacking identical flat sheets on top of one another!

Property

For a rectangular prism, the volume ($V$) is the product of its length ($l$), width ($w$), and height ($h$).

Examples

• A classroom is 35 feet long, 25 feet wide, and 10 feet high. $V = lwh = (35 \text{ ft})(25 \text{ ft})(10 \text{ ft}) = 8750 \text{ ft}^3$.
• A shoebox has dimensions of 14 inches by 8 inches by 5 inches. $V = lwh = (14 \text{ in})(8 \text{ in})(5 \text{ in}) = 560 \text{ in}^3$.

Explanation

This is a special version of the prism formula made just for boxy shapes! Since the base of a rectangular prism is a rectangle, its area ($B$) is simply length times width ($l \times w$). So, we can swap $B$ for $lw$ to get a fast and easy formula for anything from a classroom to a cereal box.

Property

The formula for the volume ($V$) of a cube is the side length ($s$) cubed.

Examples

• A 3-foot cube of ice has a volume of $V = s^3 = (3 \text{ ft})^3 = 27 \text{ ft}^3$.
• A small box is a cube with edges that are 10 cm long. $V = s^3 = (10 \text{ cm})^3 = 1000 \text{ cm}^3$.
• A single die has an edge length of 2 cm. $V = s^3 = (2 \text{ cm})^3 = 8 \text{ cm}^3$.

Explanation

A cube is the simplest 3D hero because all its sides are equal! Instead of multiplying length, width, and height separately, you just take the length of one side ($s$) and multiply it by itself three times. This awesome shortcut makes finding the volume of any cube, from a tiny sugar cube to a giant block of ice, incredibly fast.