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

• (0) Dimension: The direction in space in which a shape can be measured.
• (1-D) One-Dimensional Shape: 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.
• (2-D) Two-Dimensional Shape: 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.
• (3-D) Three-Dimensional Shape: 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 meters (m) and a width of 25 meters (m). 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(15m) + 2(25m) &\color{red}\small\text{Substitute given terms}\\\\ P &= 30m+ 50m &\color{red}\small\text{Multiplication property}\\\\ P &= 80m &\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 \({\text{cm}}^3\).
   )