Introduction

In this lesson, you will learn about units and dimensions. Units and dimensions are used all around you and can be a part of your everyday life. You will learn the difference between one dimensional, two dimensional, and three-dimensional shapes. You will also learn how to determine the perimeter, area, and volume of specific figures, and the real-world application of units and dimensions.

This video illustrates the lesson material below. Watching the video is optional.

• Units and Dimensions (08:45 mins) | Transcript

Units and Dimensions

Figure 1

• Dimension: The direction in space in which a shape can be measured.
• One-Dimensional Shape (1-D): A shape that has 1-D, a length. This means that it only consists of a length between two points. Think of a straight line, or even a curvy line drawn on a piece of paper. There is no height or width to the line, and is therefore one-dimensional. Whatever the length of the line, it only measures in one direction.
• Two-Dimensional Shape(2-D): A shape that has 2-D, a length and a width. The second dimension runs in another direction than the first, creating an actual shape. Think of a square drawn on a piece of paper, it has a length and a width, running in two directions.
• Three-Dimensional Shape (3-D): A shape that has 3-D, a length, a width, and a height. Each dimension moves in a separate direction, creating a 3-D shape. Think of a cube in front of you, it has a height, a length, and a width, running in three directions.

Measurements With Units and Dimensions

Example 2 (2-D Shape)
A rectangle of grass has a length of 15 feet (ft) and a width of 10 feet (ft). What is the perimeter (the boundary) of the shape?

Figure 2

\begin{align*}P &= 2l + 2w &\color{red}\small\text{Formula for perimeter of a rectangle}\\\\ P &= 2(15ft) + 2(25ft) &\color{red}\small\text{Substitute given terms}\\\\ P &= 30ft+ 50ft &\color{red}\small\text{Multiplication property}\\\\ P &= 80ft &\color{red}\small\text{Addition property}\\\\ \end{align*}

Example 3 (2-D Shape)
A rectangle has a length of 5m and a width of 10m. What is the area (all the inside space) of the shape?

\begin{align*}A &=(l)(w) &\color{red}\small\text{Formula for area of a rectangle}\\\\ A &= (5m)(10m) &\color{red}\small\text{Substitute given terms}\\\\ A &= 50m^2 &\color{red}\small\text{Multiplication property}\\\\ \end{align*}

The area is \(50\; square\; meters\). The unit for area changes to units squared because of the rule of exponents. This may be represented by the squared symbol or the written words (specific units) squared. 

Example 4 (3-D Shape)
A cube has the following measurements, a length of 5m, a width of 10m, and a height of 15m. What is the volume (all the inside space) of the 3-D shape?

\begin{align*}V &=(l)(w)(h) &\color{red}\small\text{Formula for volume of a rectangle}\\\\ A &= (5m)(10m)(15m) &\color{red}\small\text{Substitute given terms}\\\\ A &= 750m^3 &\color{red}\small\text{Multiplication property}\\\\ \end{align*}

The unit for volume changes to units cubed because of the rule of exponents. This may be represented by the cubed symbol \((^3)\), or the written words (specific units) cubed.

Things to Remember

• 1-D: One directional measurement is in units without any exponents.
  
  • Length is measured in units.
    
    • \( Perimeter = (2\times Length) + (2\times Width) \)
• 2-D: Two directional measurements are in \(units^2\), or square units, or units squared.
  
  • Area is measured in units squared.
    
    • \( Area = Length \times Width \)
• 3-D: Three directional measurements are in \(units^3\), or cubic units, or units cubed.
  
  • Volume is measured in units cubed.
    
    • \( Volume = Length \times Width \times Height \)

Practice Problems

1. A rectangular postage stamp has a length of 21 mm and a width of 24 mm. Find the units for the perimeter of the postage stamp. (
   Solution

   x

   Solution: Perimeter deals with only one dimension (distance or length around the outside boundary), so the answer will be millimeters (mm).
   )
2. A large piece of land is rectangular in shape and has a length of 32 miles and a width of 18 miles. Find the units for the perimeter of this piece of land. (
   Solution

   x

   Solution: Perimeter deals with only one dimension (distance or length around the outside boundary), so the answer will be miles (mi).
   )
3. A rectangular room has a length of 10 m and a width of 8 m. Find the units for the area of the room. (
   Solution

   x

   Solution: Area deals with two dimensions, or two directional measurements (length × width), so the answer will be meters squared or \({\text{m}}^2\).
   )
4. A rectangular portrait measures 16 in by 12 in. Find the units for the area of the portrait. (
   Solution

   x

   Solution: Area deals with two dimensions, or two directional measurements (length × width), so the answer will be inches squared or \({\text{in}}^2\).
   )
5. A rectangular swimming pool has a length of 16 ft, a width of 12 ft, and a depth of 6 ft. Find the units for the volume of the swimming pool. (
   Solution

   x

   Solution: Volume deals with three dimensions, or three directional measurements (length × width × height), so the answer will be feet cubed or \({\text{ft}}^3\).
   )
6. A pizza box has a square top with two adjacent sides, both measuring 33 cm. The pizza box also has a depth of 5 cm. Find the units for the volume of the pizza box. (
   Solution

   x

   Solution: Volume deals with three dimensions, or three directional measurements (length × width × height), so the answer will be centimeters cubed or \({\text{cm}}^3\).
   )

Need More Help?

1. Study other Math Lessons in the Resource Center.
2. Visit the Online Tutoring Resources in the Resource Center.
3. Contact your Instructor.
4. If you still need help, Schedule a Tutor.