The conversion factors for length can help us make the conversion factors for area if we understand how they are related to each other.
We know that 1 km = 1000m.
What is the relationship between \(\text{km}^{2}\: \text{and}\: \text{m}^{2}\) ?
Let's look at a square that is 1 km by 1 km.
We know that the area of the square is found by multiplying length × width.
The area of this square is \(1\:\text{km}\: \times \: 1\:\text{km}=1\:\text{km}^{2}\).
Since \(1\:\text{km}=1000\:\text{m}\), This is also the same as \(1000\:\text{m}\: \times \: 1000\:\text{m}=1,000,000\:\text{m}^{2}\).
So with these two pieces of information, we know \(1\:\text{km}^{2}=1,000,000\:\text{m}^{2}\).
We can now use this new conversion factor to convert between \(\text{km}^{2}\:\text{and}\:\text{m}^{2}\).
The same principle applies to finding the conversion factors between other types of square units.
For example, we can find the conversion factor between \(\text{feet}^{2}\:\text{and}\:\text{inches}^{2}\) the same way.
One square foot is a square with sides that are 1 foot long.
Since 1 foot = 12 inches we can also say this same square is 12 inches by 12 inches.
We know that the area of the square is found by multiplying \(\text{length}\: \times\: \text{width}\).
The area of this square is \(1\:\text{foot}\:\times\:1\:\text{foot}=1\:\text{foot}^{2}\).
This is also the same as \(12\:\text{inches}\:\times\:12\:\text{inches}=(12\:\text{inches})^{2}=144\:\text{inches}^{2}\).
So with these two pieces of information, we know \(1\:\text{foot}^{2}=144\:\text{inches}^{2}\).
The following video shows and explains a more complicated example of converting between different areas. It’s most important to remember that when we convert between units squared, we have to square the conversion factors.
Meet Mary and Bill. 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 \(\text{kilometers}^{2}\:\text{or}\:\text{km}^{2}\).
Mary knows that \(1\:\text{acre}=43560\:\text{feet}^{2}\).
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 \(\text{km}^{2}\).
The first thing they do is start with what they know. They know the land is 0.75 acres.
They want to get to \(\text{km}^{2}\).
The first conversion factor they use converts acres to square feet or \(\text{ft}^{2}\).
The next conversion factor that they use converts from \(\text{ft}^{2}\) to \(\text{m}^{2}\). They find it by squaring the conversion factor between feet and meters.
Finding the conversion factor this way is the same as finding it by calculating the area of the square with these dimensions. (Pretend the image below is a square.)
This allows them to convert from \(\text{ft}^{2}\) to \(\text{m}^{2}\). But they still need to get to \(\text{km}^{2}\). Thankfully, Bill knew that 1 km = 1000 m.
This helps them figure out the conversion factor between \(\text{m}^{2}\) and \(\text{km}^{2}\) in the same way that they did before.
\(1\:\text{km}^{2}=1,000,000\:\text{m}^{2}\)
They use this last conversion factor to cancel out \(\text{m}^{2}\) and finally arrive at \(\text{km}^{2}\).
They use the zig-zag method to do the calculation.
Now that both Bill and Mary know how big the piece of land is, they can agree on a fair price for the land.
Practice Problems
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.
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. \((10\:\text{mm})^{2}=(1\:\text{cm})^{2}\)
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.
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 miles = 1.60934 kilometers. Round your answer to the nearest whole number.
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 yard = 3 feet.
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 foot = 12 inches.
In the problem, we are given 145 \(\text{ft}^{2}\), so the units we have are feet squared (\(\text{ft}^{2}\))
Step 2: Figure out what units we want.
The problem asks us to convert feet squared (\(\text{ft}^{2}\)) into meters squared (\(\text{m}^{2}\)), so we want, meters squared (\(\text{m}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 1 ft = 0.3048 m; however, since we have feet squared and want meters squared we need to square both sides of our conversion factor so that we 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.
We know that we have feet squared (\(\text{ft}^{2}\)) and we want meters squared (\(\text{m}^{2}\)). We want \(\text{ft}^{2}\) to cancel out, so we will put \(\text{ft}^{2}\) in the denominator and \(\text{m}^{2}\) in the numerator of our conversion factor.
In the problem, we are given 550 \(\text{mm}^{2}\), so the units we have are millimeters squared (\(\text{mm}^{2}\))
Step 2: Figure out what units we want.
The problem asks us to convert millimeters squared (\(\text{mm}^{2}\)) into centimeters squared(\(\text{cm}^{2}\)), so we want centimeters squared (\(\text{cm}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 10 mm = 1 cm; however, since we have millimeters squared (\(\text{mm}^{2}\)) and want centimeters squared (\(\text{cm}^{2}\)), we need to square both sides of our conversion factor so that we 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.
We know that we have millimeters squared (\(\text{mm}^{2}\)) and we want centimeters squared (\(\text{cm}^{2}\)). We want \(\text{mm}^{2}\) to cancel out, so we will put \(\text{mm}^{2}\) in the denominator and \(\text{cm}^{2}\) in the numerator of our conversion factor.
In the problem, we are given 1440 \(\text{in}^{2}\), so the units we have are inches squared (\(\text{in}^{2}\))
Step 2: Figure out what units we want.
The problem asks us to convert inches squared (\(\text{in}^{2}\)) into centimeters squared (\(\text{cm}^{2}\)), so we want centimeters squared (\(\text{cm}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 1 in = 2.54 cm; however, since we have inches squared (\(\text{in}^{2}\)) and want centimeters squared (\(\text{cm}^{2}\)), we need to square both sides of our conversion factor so that we 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.
We know that we have inches squared (\(\text{in}^{2}\)) and we want centimeters squared (\(\text{cm}^{2}\)). We want \(\text{in}^{2}\) to cancel out, so we will put \(\text{in}^{2}\) in the denominator and \(\text{cm}^{2}\) in the numerator of our conversion factor.
In the problem, we are given information to find the area of the ranch. We know that the length is 31 miles and the width is 26 miles. Multiplying length × width, we get 31 mi × 26 mi = (31 × 26) × (mi × mi) = 806 \(\text{mi}^{2}\), so the units we have are miles squared (\(\text{mi}^{2}\)).
Step 2: Figure out what units we want.
The problem asks us to convert miles squared (\(\text{mi}^{2}\)) into kilometers squared (\(\text{km}^{2}\)), so we want kilometers squared (\(\text{km}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 1 mi = 1.60934 km; however, since we have miles squared (\(\text{mi}^{2}\)) and want kilometers squared (\(\text{km}^{2}\)), we need to square both sides of our conversion factor so that we 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.
We know that we have miles squared (\(\text{mi}^{2}\)) and we want kilometers squared (\(\text{km}^{2}\)). We want \(\text{mi}^{2}\) to cancel out, so we will put \(\text{mi}^{2}\) in the denominator and \(\text{km}^{2}\) in the numerator of our conversion factor.
In the problem, we are given information to find the surface area of the tennis court. We know that the length is 78 feet and the width is 27 feet. Multiplying length × width, we have 78 ft × 27 ft = (78 × 27) × (ft × ft) = 2106 \(\text{ft}^{2}\), so the units we have are feet squared (\(\text{ft}^{2}\)).
Step 2: Figure out what units we want.
The problem asks us to convert feet squared (\(\text{ft}^{2}\)) into yards squared (\(\text{yd}^{2}\)), so we want yards squared (\(\text{yd}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 1 yd = 3 ft; however, since we have feet squared (\(\text{ft}^{2}\)) and want yards squared (\(\text{yd}^{2}\)), we need to square both sides of our conversion factor so that we 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.
We know that we have feet squared (\(\text{ft}^{2}\)) and we want yards squared (\(\text{yd}^{2}\)). We want \(\text{ft}^{2}\) to cancel out, so we will put \(\text{ft}^{2}\) in the denominator and \(\text{yd}^{2}\) in the numerator of our conversion factor.
In the problem, we are given 112 \(\text{ft}^{2}\), so the units we have are feet squared (\(\text{ft}^{2}\)).
Step 2: Figure out what units we want.
The problem asks us to convert feet squared (\(\text{ft}^{2}\)) into inches squared (\(\text{in}^{2}\)), so we want inches squared (\(\text{in}^{2}\)).
Step 3: Find conversion factors that will help get the units we want.
We are given the conversion factor 1 ft = 12 in; however, since we have feet squared (\(\text{ft}^{2}\)) and want inches squared (\(\text{in}^{2}\)), we need to square both sides of our conversion factor so that we 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.
We know that we have feet squared (\(\text{ft}^{2}\)) and we want inches squared (\(\text{in}^{2}\)). We want \(\text{ft}^{2}\) to cancel out, so we will put \(\text{ft}^{2}\) in the denominator and \(\text{in}^{2}\) in the numerator of our conversion factor.
