Lesson 3.1: Volume and Surface Area

New Concept

This lesson introduces two key measurements for 3D objects: surface area, the total area of all outer faces, and volume, the space enclosed within. You will learn to calculate both for boxes and cubes, distinguishing their unique properties and units.

What’s next

Get ready to apply these concepts. You'll work through interactive practice cards and challenge problems to master calculating surface area and volume for various 3D shapes.

Property

When we calculate the sum of the areas of the outside faces of a three-dimensional object, it is called the surface area of the object. For a box with length $l$, width $w$, and height $h$, the surface area is the sum of the areas of its six faces: front, back, top, bottom, left, and right.

Examples

• A rectangular box has dimensions 5 cm, 3 cm, and 2 cm. Its surface area is $2(5 \times 3) + 2(5 \times 2) + 2(3 \times 2) = 30 + 20 + 12 = 62$ square cm.

• A book measures 8 inches wide, 10 inches long, and 1 inch thick. Its surface area is $2(8 \times 10) + 2(8 \times 1) + 2(10 \times 1) = 160 + 16 + 20 = 196$ square inches.

Property

A cube is a box whose length, width, and height are all equal. This means that all six faces are identical squares. If the side length of a cube is $s$, the area of each face is $s^2$. The total surface area of the cube is $6 \times s^2$.

Examples

• A wooden block is a cube with a side length of 4 inches. The area of one face is $4^2 = 16$ square inches. The total surface area is $6 \times 16 = 96$ square inches.

• A sugar cube has sides of 1 cm. Its surface area is $6 \times (1 \text{ cm})^2 = 6$ square cm.

Property

We use cubic units to measure the volume or amount of space inside a three-dimensional object. For a box with dimensions length $l$, width $w$, and height $h$, the volume is found by multiplying the three dimensions.

Examples

• An aquarium is 3 feet long, 1.5 feet wide, and 2 feet tall. Its volume is $3 \times 1.5 \times 2 = 9$ cubic feet.

• A shoebox has dimensions of 14 inches by 8 inches by 5 inches. Its volume is $14 \times 8 \times 5 = 560$ cubic inches.

Property

Surface area measures the area of the outside surface of a three-dimensional object. Volume measures the amount of space inside the object. Surface area is measured in square units, and volume is measured in cubic units.

Examples

• A cube with a side of 1 inch has a surface area of $6 \times 1^2 = 6$ square inches and a volume of $1^3 = 1$ cubic inch.

• A cube with a side of 2 meters has a surface area of $6 \times 2^2 = 24$ square meters and a volume of $2^3 = 8$ cubic meters.