Introduction

In this lesson, you will learn how to convert between units when calculating area, and you will practice using unit conversions to find the area. You can also do unit conversions of areas, where the units are squared.

These videos illustrate the lesson material below. Watching the videos is optional.

• Unit Conversions for Area (04:16 mins) | Transcript
• Example of Unit Conversions with Area (09:01 mins) | Transcript

Unit Conversions for Area

You probably know that:

\begin{align*} 1\;kilometer = 1000\;meters \end{align*}

But what is the conversion factor between:

\begin{align*} kilometers^{2} \rightarrow \;? \;meters^{2} \end{align*}

Think about what it means to go from kilometers to \(kilometers^{2}\), or \(km^{2}\), and from meters to \(meters^{2}\), or \(m^{2}\).

Example 1
Find the conversion factor to convert a measurement from kilometers squared to meters squared.

The square in Figure 1 is 1 kilometer in length. The conversion factor of kilometers to meters is 1 kilometer equals 1000 meters.

Figure 1

The area of a square is the same as the length multiplied by the width, so the area of this square is
\begin{align*} 1km\times1km=1km^{2}\end{align*}

It is also equal to:
\begin{align*} 1000m\times1000m=1,000,000m^{2}\end{align*}

Figure 2

Using what you know about what it means to become a square unit, you can find the conversion between \(km^{2}\) and \(m^{2}\):
\begin{align*} 1km^{2}=1,000,000m^{2}\end{align*}

This is a conversion factor that you can use if you need to convert a measurement from kilometers squared to meters squared.

Example 2
Find the conversion factor to convert a measurement from feet squared to inches squared. 1 foot, or 1ft, is equal to 12 inches, or 12in.

\begin{align*} 1 ft = 12 in \end{align*}

Figure 3

The area of the square in Figure 3 is 1 square foot. This means that each side of the square is 1 foot long.

\begin{align*} 1 ft\times1 ft=1ft^{2} \end{align*}
If \(1ft=12in\), that means the area of the square is also equal to:
\begin{align*} 12in\times12in=144in^{2}\end{align*}

Figure 4

There are \(144in^{2}\) within \(1ft^{2}\), so the area conversion between square feet and square inches is:

\begin{align*} 1ft^{2}=144in^{2} \end{align*}

Example 3

In this example, Mary owns a piece of property that Bill wants to buy. Mary uses acres to measure her land. She had the land measured by a surveyor who said the land is 0.75 acres. Bill uses square kilometers to measure land. In order for Bill and Mary to agree on a price for the land, they need to figure out the size of the land in \( kilometers^2\; or\; km^2 \).

Figure 5

• Mary knows that \( 1\; acre = 43560\; feet^2 \).
• Bill knows that \(1\; m = 3.281 \;ft\).

Bill and Mary work together to figure out how big this piece of land is in square kilometers or \( km^2 \).
The first thing they do is start with what they know. They know the land is 0.75 acres. They want to get to \( km^2 \). The first conversion factor they use converts acres to square feet or \( ft^2 \).

\begin{align*} 0.75\:\text{acres}\times\dfrac{43560\:\text{ft}^{2}}{1 \:\text{acre}}\end{align*}

The next conversion factor that they use converts from \( ft^2 \) to \( m^2 \). They find it by squaring the conversion factor between feet and meters.

\begin{align*} \frac{\left ( 1\:\text{m} \right )^{2}}{\left ( 3.281\:\text{ft} \right )^{2}} \end{align*}

Squaring the numerator and denominator gives them the following conversion factor.

\begin{align*} \frac{\left ( 1\:\text{m} \right )^{2}}{\left ( 3.281\:\text{ft} \right )^{2}} &= \frac{1\:\text{m} ^{2}}{10.765\:\text{ft}^{2}} \\\\ 1\:\text{m}^{2} &=10.765\:\text{ft}^{2}\end{align*}

Finding the conversion factor this way is the same as finding it by calculating the area of the square with these dimensions. They will use conversion factors to find how many kilometers in 0.75 acres.

\begin{align*} &0.75\;acres =?\; km^2 & \color{red}\small\text{Convert acres to \(km^2\)}\\\\ &0.75\;{\color{red}\cancel{acres}} \times \frac{43560\;ft^2}{1\;\color{red}\cancel{acre}} & \color{red}\small\text{Use conversion factor to cancel acre}\\\\ &0.75\;{\color{red}\cancel{acres}} \times \frac{43560\;\color{blue}\cancel{ft^{2}}}{1\;\color{red}\cancel{acre}} \times \frac{1\;m^2}{10.765\;\color{blue}\cancel{ft^{2}}} & \color{red}\small\text{Use the conversion factor to cancel \(ft^2\)}\\\\ \end{align*}

This allows them to convert from acres to \( m^2 \). But they still need to get to \( km^2 \). Thankfully, Bill knew that \(1\; km = 1000\; m\).

This helps them figure out the conversion factor between \( km^2 \) and \( m^2 \) in the same way that they did before.

\begin{align*} 1\; km^2 = 1,000,000\; m^2 \end{align*}

They use this last conversion factor to cancel out \( m^2 \) and finally arrive at \( km^2 \).

\begin{align*}\displaystyle 0.75\:\cancel{{\color{Red} \text{acres}}}\times\frac{43560\:\cancel{ {\color{blue}\text{ft}^{2}}}}{1 \:\cancel{{\color{Red} \text{acre}}}}\times\frac{1\:\cancel{{\color{green} \text{m} ^{2}}}}{10.765\:\cancel{{\color{blue} \text{ft}^{2}}}}\times \frac{1\:\text{km}^{2}}{1,000,000\:\cancel{{\color{blue} \text{m}^{2}}}}\end{align*}

They use the zig-zag method to do the calculation.

\begin{align*}0.75\:\div\:1\:\times43560\:\div\:1\:\times\:1\:\div\:10.765\: \times\:1\:\div\:1,000,000=0.003\:\text{km}^{2}\end{align*}

Or they can perform this calculation by using a calculator.

\begin{align*} &\frac{0.75 \times 43560 \times 1\times 1\;km^2}{1 \times10.765 \times 1000000} = \frac{32670\;km^2}{10764000} = 0.03\;km \end{align*}

So, \(0.75\:\text{acres}^{2}=0.003\:\text{km}^{2}\).

Now that both Bill and Mary know how big the piece of land is, they can agree on a fair price for the land.

Things to Remember

• Use units of conversion to move from one unit of measurement to another.
• When multiplying many fractions, there are two methods that can be used
  
  • Method 1: Multiply the numerator and denominator across first, and then simply or
  • Method 2: Simplify the fractions first by cross canceling and then multiply numerator and denominator across.

Practice Problems

1. The surface area of a small bedroom is \(145\:\text{ft}^{2}\). Use the fact that \(1\; ft = 0.3048\; m\) to convert this area to \(\text{m}^{2}\). Round to the nearest tenth. (
   Solution

   x

   Solution: \(13.5\text{m}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given 145 \(\text{ft}^{2}\), so the units you have are feet squared (\(\text{ft}^{2}\)).

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert feet squared (\(\text{ft}^{2}\)) into meters squared (\(\text{m}^{2}\)), so you want meters squared (\(\text{m}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor \(1 ft = 0.3048 m\); however, since you have feet squared and want meters squared, you need to square both sides of the conversion factor so that you are dealing with units in the same dimension.

   \((1\:\text{ft})^{2}=(0.3048\:\text{m})^{2}\)
   
   \(1\:\text{ft}^{2}=0.0929\:\text{m}^{2}\)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have feet squared (\(\text{ft}^{2}\)) and you want meters squared (\(\text{m}^{2}\)). You want \(\text{ft}^{2}\) to cancel out, so you will put \(\text{ft}^{2}\) in the denominator and \(\text{m}^{2}\) in the numerator of the conversion factor.

   \(\displaystyle\frac{145\:\text{foot}^{2}}{1}\times \frac{0.0929\:\text{meter}^{2}}{1\:\text{foot}^{2}}\)
   
   The \(\text{ft}^{2}\) will cancel, and you can use the zig-zag method to find the answer. (Calculate from left to right.)
   
   \(145\div 1\times 0.0929 \:\text{m}^{2}\div 1=13.5\:\text{m}^{2}\)
   )
2. The surface area of a postage stamp is \(550\:\text{mm}^{2}\). Use the fact that \(10\; mm = 1\; cm\) to convert this area to \(\text{cm}^{2}\). Round to the nearest tenth. \((10\:\text{mm})^{2}=(1\:\text{cm})^{2}\) (
   Solution

   x

   Solution: \(5.5 \text{cm}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given 550 \(\text{mm}^{2}\), so the units you have are millimeters squared (\(\text{mm}^{2}\)).

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert millimeters squared (\(\text{mm}^{2}\)) into centimeters squared(\(\text{cm}^{2}\)), so you want centimeters squared (\(\text{cm}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor 10 mm = 1 cm; however, since you have millimeters squared (\(\text{mm}^{2}\)) and want centimeters squared (\(\text{cm}^{2}\)), you need to square both sides of the conversion factor so that you are dealing with units in the same dimension:

   \((10\:\text{mm})^{2}=(1\:\text{cm})^{2}\)
   
   \(100\:\text{mm}^{2}=1\:\text{cm}^{2}\)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have millimeters squared (\(\text{mm}^{2}\)) and you want centimeters squared (\(\text{cm}^{2}\)). You want \(\text{mm}^{2}\) to cancel out, so you will put \(\text{mm}^{2}\) in the denominator and \(\text{cm}^{2}\) in the numerator of the conversion factor.

   \(\displaystyle\frac{550\:\text{mm}^{2}}{1}\times \frac{1\:\text{cm}^{2}}{100\:\text{mm}^{2}}\)
   
   The \(\text{mm}^{2}\) will cancel, and you can use the zig-zag method to find the answer. (Calculate from left to right.)
   
   \(550\div 1\times 1\:\text{cm}^{2}\div 100=5.5\:\text{cm}^{2}\)
   )
3. The surface area of a tabletop is \(1440\:\text{in}^{2}\). Use the fact that \(1\; in = 2.54\; cm\) to convert this area to \(\text{cm}^{2}\). Round to the nearest tenth. (
   Solution

   x

   Solution: \(9290.3 \text{cm}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given 1440 \(\text{in}^{2}\), so the units you have are inches squared (\(\text{in}^{2}\))

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert inches squared (\(\text{in}^{2}\)) into centimeters squared (\(\text{cm}^{2}\)), so you want centimeters squared (\(\text{cm}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor 1 in = 2.54 cm; however, since you have inches squared (\(\text{in}^{2}\)) and want centimeters squared (\(\text{cm}^{2}\)), you need to square both sides of the conversion factor so that you are dealing with units in the same dimension:

   \((1\:\text{in})^{2}=(2.54\:\text{cm})^{2}\)
   
   \(1\:\text{in}^{2}=6.4516\:\text{cm}^{2}\)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have inches squared (\(\text{in}^{2}\)) and you want centimeters squared (\(\text{cm}^{2}\)). You want \(\text{in}^{2}\) to cancel out, so you will put \(\text{in}^{2}\) in the denominator and \(\text{cm}^{2}\) in the numerator of the conversion factor.

   \(\dfrac{1440\:\text{in}^{2}}{1}\times \dfrac{6.4516\:\text{cm}^{2}}{1\:\text{in}^{2}}\)
   
   The \(\text{in}^{2}\) will cancel, and you can use the zig-zag method to find the answer.
   
   \(1440\div 1\times 6.4516\:\text{cm}^{2}\div 1=9290.3\:\text{cm}^{2}\)
   )
4. A large ranch has dimensions that are rectangular in shape with a length of 31 miles and a width of 26 miles. Find the area of the ranch in miles squared (\(\text{mi}^{2}\)) and then convert this to kilometers squared (\(\text{km}^{2}\)) using the fact that \(1\; mi = 1.60934\; km\). Round to the nearest whole number. (
   Solution

   x

   Solution: \(2088 \text{km}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given information to find the area of the ranch. You know that the length is 31 miles and the width is 26 miles. Multiplying \(length \times width\), you get \(31 mi \times 26 mi = (31 \times 26) \times (mi \times mi) = 806 \text{mi}^{2}\), so the units you have are miles squared (\(\text{mi}^{2}\)).

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert miles squared (\(\text{mi}^{2}\)) into kilometers squared (\(\text{km}^{2}\)), so you want kilometers squared (\(\text{km}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor \(1 mi = 1.60934 km\); however, since you have miles squared (\(\text{mi}^{2}\)) and want kilometers squared (\(\text{km}^{2}\)), you need to square both sides of the conversion factor so that you are dealing with units in the same dimension:

   \((1\:\text{mi})^{2}=(1.60934\:\text{km})^{2}\)
   
   \(1\:\text{mi}^{2}=2.59\:\text{km}^{2}\) (This was rounded to the nearest hundredth.)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have miles squared (\(\text{mi}^{2}\)) and you want kilometers squared \((\text{km}^{2})\). You want (\(\text{mi}^{2}\)) to cancel out, so you will put (\(\text{mi}^{2}\)) in the denominator and \((\text{km}^{2}\)) in the numerator of the conversion factor.

   \(\dfrac{806\:\text{mi}^{2}}{1}\times \dfrac{2.59\:\text{km}^{2}}{1\:\text{mi}^{2}}\)
   
   The \(\text{mi}^{2}\) will cancel, and you can use the zig-zag method to find the answer, calculating from left to right.
   
   \(806\div 1\times 2.59\:\text{km}^{2}\div 1=2088\:\text{km}^{2}\)
   )
5. Susan knows that a tennis court has a length of 78 feet and a width of 27 feet. She needs to know the surface area of the court in yards squared (\(\text{yd}^{2}\)). Find the surface area of the tennis court in feet squared (\(\text{ft}^{2}\)), and then convert this to yards squared (\(\text{yd}^{2}\)). Use the fact that \(1\; yd = 3\; ft\). (
   Solution

   x

   Solution: \(234 \text{yd}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given information to find the surface area of the tennis court. You know that the length is 78 feet and the width is 27 feet. Multiplying \(length \times width\), you have \(78\; ft \times 27\; ft = (78 \times 27) \times (ft \times ft) = 2106 \text{ft}^{2}\), so the units you have are feet squared (\(\text{ft}^{2}\)).

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert feet squared (\(\text{ft}^{2}\)) into yards squared (\(\text{yd}^{2}\)), so you want yards squared (\(\text{yd}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor \(1\; yd = 3\; ft\); however, since you have feet squared (\(\text{ft}^{2}\)) and want yards squared (\(\text{yd}^{2}\)), you need to square both sides of the conversion factor so that you are dealing with units in the same dimension:

   \((1\:\text{yd})^{2}=(3\:\text{ft})^{2}\)
   
   \(1\:\text{yd}^{2}=9\:\text{ft}^{2}\)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have feet squared (\(\text{ft}^{2}\)) and you want yards squared (\(\text{yd}^{2}\)). You want \(\text{ft}^{2}\) to cancel out, so you will put \(\text{ft}^{2}\) in the denominator and \(\text{yd}^{2}\) in the numerator of the conversion factor.

   \(\dfrac{2106 \:\text{ft}^{2}}{1}\times \dfrac{1\:\text{yd}^{2}}{9\:\text{ft}^{2}}\)
   
   The \(\text{ft}^{2}\) will cancel, and you can use the zig-zag method to find the answer, calculating from left to right.
   
   \(2106\div 1\times 1\:\text{yd}^{2}\div 9=234\:\text{yds}^{2}\)
   )
6. A garage door has a surface area of \(112\:\text{ft}^{2}\). Find the surface area of the garage in inches squared (\(\text{in}^{2}\)) using the fact that \(1\; ft = 12\; in\). (
   Solution

   x

   Solution: \(16128 \text{in}^{2}\)
   Details:
   Step 1: Find the units you have.
   

   • In the problem, you are given 112 \(\text{ft}^{2}\), so the units you have are feet squared (\(\text{ft}^{2}\)).

   Step 2: Figure out what units you want.
   

   • The problem asks you to convert feet squared (\(\text{ft}^{2}\)) into inches squared (\(\text{in}^{2}\)), so you want inches squared (\(\text{in}^{2}\)).

   Step 3: Find conversion factors that will help get the units you want.
   

   • You are given the conversion factor \(1 ft = 12 in\); however, since you have feet squared (\(\text{ft}^{2}\)) and want inches squared (\(\text{in}^{2}\)), you need to square both sides of the conversion factor so that you are dealing with units in the same dimension:

   \((1\:\text{ft})^{2}=(12\:\text{in})^{2}\)
   
   \(1\:\text{ft}^{2}=144\:\text{in}^{2}\)
   
   Step 4: Arrange conversion factors so unwanted units cancel out.
   

   • You know that you have feet squared (\(\text{ft}^{2}\)) and you want inches squared (\(\text{in}^{2}\)). You want \(\text{ft}^{2}\) to cancel out, so you will put \(\text{ft}^{2}\) in the denominator and \(\text{in}^{2}\) in the numerator of the conversion factor.

   \(\dfrac{112\:\text{ft}^{2}}{1}\times \dfrac{144\:\text{in}^{2}}{1\:\text{ft}^{2}}\)
   
   The \(\text{ft}^{2}\) will cancel, and you can use the zig-zag method to find the answer, calculating from left to right.
   
   \(112\div 1\times 144\:\text{in}^{2}\div 1=16128\:\text{in}^{2}\)
   )

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