9.2: Measuring Area
- Page ID
- 129628
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Identify reasonable values for area applications.
- Convert units of measures of area.
- Solve application problems involving area.
Area is the size of a surface. It could be a piece of land, a rug, a wall, or any other two-dimensional surface with attributes that can be measured in the metric unit for distance-meters. Determining the area of a surface is important to many everyday activities. For example, when purchasing paint, you’ll need to know how many square units of surface area need to be painted to determine how much paint to buy.
Square units indicate that two measures in the same units have been multiplied together. For example, to find the area of a rectangle, multiply the length units and the width units to determine the area in square units.
\[1 \mathrm{~cm} \times 1 \mathrm{~cm} =1\mathrm{~cm}^2\]
Note that to accurately calculate area, each of the measures being multiplied must be of the same units. For example, to find an area in square centimeters, both length measures (length and width) must be in centimeters.
The formula used to determine area depends on the shape of that surface. Here we will limit our discussions to the area of rectangular-shaped objects like the one in Figure 9.6. Given this limitation, the basic formula for area is:
Reasonable Values for Area
Because area is determined by multiplying two lengths, the magnitude of difference between different square units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter ( \(\mathrm{m}^2\) ) is \(100 \times 100\), or 10, 000 times greater in area than a square centimeter \(\left(\mathrm{cm}^2\right)\). The relationships between benchmark metric area units are shown in the following table.
|
Units |
Relationship |
Conversion Rate |
| \(\mathrm{km}^2\) to \(\mathrm{m}^2\) | \(\begin{aligned} \mathrm{km} \times \mathrm{km} & =\mathrm{km}^2 \\ 1 \mathrm{~km} & =1,000 \mathrm{~m} \\ 1,000 \mathrm{~m} \times 1,000 \mathrm{~m} & =1,000,000 \mathrm{~m}^2\end{aligned}\) | \(1 \mathrm{~km}^2=1,000,000 \mathrm{~m}^2\) |
| \(\mathrm{m}^2\) to \(\mathrm{cm}^2\) | \(\begin{aligned} \mathrm{m} \times \mathrm{m} & =\mathrm{m}^2 \\ 1 \mathrm{~m} & =100 \mathrm{~cm} \\ 100 \mathrm{~cm} \times 100 \mathrm{~cm} & =10,000 \mathrm{~cm}^2\end{aligned}\) | \(1 \mathrm{~m}^2=10,000 \mathrm{~cm}^2\) |
| \(\mathrm{cm}^2\) to \(\mathrm{mm}^2\) | \(\begin{aligned} \mathrm{cm} \times \mathrm{cm} & =\mathrm{cm}^2 \\ 1 \mathrm{~cm} & =10 \mathrm{~mm} \\ 10 \mathrm{~mm} \times 10 \mathrm{~mm} & =100 \mathrm{~mm}^2\end{aligned}\) | \(1 \mathrm{~cm}^2=100 \mathrm{~mm}^2\) |
An essential understanding of metric area is to identify reasonable values for area. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given area is reasonable for the situation.
Which unit of measure is most reasonable to describe the area of a sheet of paper: \(\mathrm{km}^2, \mathrm{~cm}^2\), or \(\mathrm{mm}^2\) ?
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In the U.S. Customary System of Measurement, the length and width of paper is usually measured in inches. In the metric system centimeters are used for measures usually expressed in inches. Thus, the most reasonable unit of measure to describe the area of a sheet of paper is square centimeters. Square kilometers is too large a unit and square millimeters is too small a unit.
Which unit of measure is most reasonable to describe the area of a forest: km2, cm2, or mm2?
You want to paint your bedroom walls. Which represents a reasonable value for the area of the walls: 100 cm2, 100 m2, or 100 km2?
- Answer
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An area of 100 cm2 is equivalent to a surface of
10 cm × 10 cm , which is much too small for the walls of a bedroom. An area of 100 km2 is equivalent to a surface of10 km × 10 km , which is much too large for the walls of a bedroom. So, a reasonable value for the area of the walls is 100 m2.
Which represents a reasonable value for the area of the top of a kitchen table: 1,800 mm2, 1,800 cm2, or \(\text{1,800 m}^2\)?
A landscaper is hired to resod a school’s football field. After measuring the length and width of the field they determine that the area of the football field is 5,350 km2. Does their calculation make sense? Explain your answer.
- Answer
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No, kilometers are used to determine longer distances, such as the distance between two points when driving. A football field is less than 1 kilometer long, so a more reasonable unit of value would be m2. An area of 5,350 km2 can be calculated using the dimensions 53.5 by 100, which are reasonable dimensions for the length and width of a football field. So, a more reasonable value for the area of the football field is 5,350 m2.
A crafter uses four letter-size sheets of paper to create a paper mosaic in the shape of a square. They decide they want to frame the paper mosaic, so they measure and determine that the area of the paper mosaic is \(\text{2,412.89 cm}^2\). Does their calculation make sense? Explain your answer.
Converting Units of Measures for Area
Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are shown in Figure 9.7.
Converting Metric Units of Area
A plot of land has an area of 237,500,000 m2. What is the area in square kilometers?
- Answer
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Use division to convert from a smaller metric area unit to a larger metric area unit. To convert from m2 to km2, divide the value of the area by 1,000,000.
The plot of land has an area of 237.5 km2.
A roll of butcher paper has an area of 1,532,900 cm2. What is the area of the butcher paper in square meters?
A plot of land has an area of 0.004046 km2. What is the area of the land in square meters?
- Answer
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Use multiplication to convert from a larger metric area unit to a smaller metric area unit. To convert from km2 to m2, multiply the value of the area by 1,000,000.
The plot of land has an area of 4,046 m2.
A bolt of fabric has an area of 136.5 m2. What is the area of the bolt of fabric in square centimeters?
A computer chip measures 10 mm by 15 mm. How many square centimeters is the computer chip?
- Answer
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Step 1: Convert the measures of the computer chip into centimeters
\[\begin{array}{l}
10 \mathrm{~mm}=1 \mathrm{~cm} \\
15 \mathrm{~mm}=1.5 \mathrm{~cm}
\end{array}\]Step 2: Use the area formula to determine the area of the chip.
\[1 \times 1.5=1.5\]
The computer chip has an area of \(1.5 \mathrm{~cm}^2\).
A piece of fabric measures 100 cm by 106 cm. What is the area of the fabric in square meters?
Solving Application Problems Involving Area
While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of the surface that are not relevant to your calculation.
One side of a commercial building is 12 meters long by 9 meters high. There is a rolling door on this side of the building that is 4 meters wide by 3 meters high. You want to refinish the side of the building, but not the door, with aluminum siding. How many square meters of aluminum siding are required to cover this side of the building?
- Answer
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Step 1: Determine the area of the side of the building.
\[12 \mathrm{~m} \times 9 \mathrm{~m}=106 \mathrm{~m}^2\]
Step 2: Determine the area of the door.
\[4 \mathrm{~m} \times 3 \mathrm{~m}=12 \mathrm{~m}^2\]
Step 3: Subtract the area of the door from the area of the side of the building.
\[106 \mathrm{~m}^2-12 \mathrm{~m}^2=92 \mathrm{~m}^2\]
So, you need to purchase \(92 \mathrm{~m}^2\) of aluminum siding.
You want to cover a garden with topsoil. The garden is 5 meters by 8 meters. There is a path in the middle of the garden that is 8 meters long and 0.75 meters wide. What is the area of the garden you need to cover with topsoil?
When calculating area, you must ensure that both distance measurements are expressed in terms of the same distance units. Sometimes you must convert one measurement before using the area formula.
A national park has a land area in the shape of a rectangle. The park measures 2.2 kilometers long by 1,250 meters wide. What is the area of the park in square kilometers?
- Answer
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Step 1: Use a conversion fraction to convert the information given in meters to kilometers.
Step 2: Multiply to find the area.
The park has an area of 2.75 km2.
An Olympic pool measures 50 meters by 2,500 centimeters. What is the surface area of the pool in square meters?
When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not relevant.
A kitchen floor has an area of 15 m2. The floor in the kitchen pantry is 100 cm by 200 cm. You want to tile the kitchen and pantry floors using the same tile. How many square meters of tile do you need to buy?
- Answer
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Step 1: Determine the area of the pantry floor in square centimeters.
Step 2: Divide the area in cm2 by the conversion factor to determine the area in m2 since the other measurement for the kitchen floor is in m2.
The area of the kitchen pantry floor is 2 m2.
Step 3: Add the two areas of the pantry and the kitchen floors together.
So, you need to buy 17 m2 of tile.
Your bedroom floor has an area of 25 m2. The living room floor measures 600 cm by 750 cm. How many square meters of carpet do you need to buy to carpet the floors in both rooms?
The metric system is the official measurement system for every country in the world except the United States, Liberia, and Myanmar. But did you know it originated in France during the French Revolution in the late 18th century? At the time there were over 250,000 different units of weights and measures in use, often determined by local customs and economies. For example, land was often measured in days, referring to the amount of land a person could work in a day.
Why the Metric System Matters
Check Your Understanding
Determine the most reasonable value for area of a bedroom wall: 12 km2, 12 m2, 12 cm2, or 12 mm2
Determine the most reasonable value for area of a city park: 1,200 km2, 1,200 m2, 1,200 cm2, or 1,200 mm2
Determine the most reasonable value for area of a kitchen table: 2.5 km2, 2.5 m2, 2.5 cm2, or 2.5 mm2
Convert the given area to the units shown.
20,000 cm2 = __________ m2
Convert the given area to the units shown.
5.7 m2 = __________ cm2
Convert the given area to the units shown.
217 cm2 = __________ mm2
A wall measures 4 m by 2 m. A doorway in the wall measures 0.5 m by 1.6 m. What is the area of the wall not taken by the door in square meters?