Lesson 6: Squares and Cubes

Property

A standard unit of area is a square with a specific side length used for consistent measurement.

• A square inch ($1 \text{ in}^2$) is a square with 1-inch sides.
• A square centimeter ($1 \text{ cm}^2$) is a square with 1-centimeter sides.

Since 1 inch is longer than 1 centimeter, a square inch is a larger unit of area: $1 \text{ in}^2 > 1 \text{ cm}^2$.

Property

Volume is a measure of how much space is inside a three-dimensional object or how much it takes to fill a container.
Volume is always measured in cubic units such as cubic inches ($\text{in}^3$), cubic feet ($\text{ft}^3$), cubic centimeters ($\text{cm}^3$), or cubic meters ($\text{m}^3$).

Examples

Property

For a cube with side length $s$:

• Volume: $V = s^3$
• Surface Area: $SA = 6s^2$

Examples

• A cube with side length $s = 4$ cm has a volume of $V = 4^3 = 64$ cm$^3$ and a surface area of $SA = 6 \cdot 4^2 = 6 \cdot 16 = 96$ cm$^2$.
• A cube with side length $s = 10$ ft has a volume of $V = 10^3 = 1000$ ft$^3$ and a surface area of $SA = 6 \cdot 10^2 = 6 \cdot 100 = 600$ ft$^2$.

Explanation

The volume of a cube is found by cubing its side length, which means multiplying the side length by itself three times. The surface area is the total area of all its faces. Since a cube has 6 identical square faces, you find the area of one face by squaring the side length ($s^2$) and then multiplying it by 6.