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Measuring Area

Identify reasonable values for area applications.

Learning Objectives

After completing this section, you should be able to:

Identify reasonable values for area applications.
Convert units of measures of area.
Solve application problems involving area.

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 \(({\text{m}}^{\text{2}})\) is \(100\times 100,\text{or}10,000\) times greater in area than a square centimeter \(({\text{cm}}^{\text{2}})\). The relationships between benchmark metric area units are shown in the following table.

Units	Relationship	Conversion Rate
\({\text{km}}^{2}\) to \({\text{m}}^{2}\)	\(\begin{array}{lll}\text{km}\times \text{km} & = & {\text{km}}^{2} \\ 1\text{km} & = & 1,000m \\ 1,000m\times 1,000m & = & 1,000,000{m}^{2}\end{array}\)	\(1{\text{km}}^{2}=1,000,000{m}^{2}\)
\({\text{m}}^{2}\) to \({\text{cm}}^{2}\)	\(\begin{array}{lll}m\times m & = & {m}^{2} \\ 1m & = & 100\text{cm} \\ 100\text{cm}\times 100\text{cm} & = & 10,000{\text{cm}}^{2}\end{array}\)	\(1{m}^{2}=10,000{\text{cm}}^{2}\)
\({\text{cm}}^{2}\) to \({\text{mm}}^{2}\)	\(\begin{array}{lll}\text{cm}\times \text{cm} & = & {\text{cm}}^{2} \\ 1\text{cm} & = & 10\text{mm} \\ 10\text{mm}\times 10\text{mm} & = & 100{\text{mm}}^{2}\end{array}\)	\(1{\text{cm}}^{2}=100{\text{mm}}^{2}\)

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.

Determining Reasonable Units for Area

Try it.

Which unit of measure is most reasonable to describe the area of a sheet of paper: \({\text{km}}^{2}\), \({\text{cm}}^{2}\), or \({\text{mm}}^{2}\)?

Solution

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.

Determining Reasonable Values for Area

Try it.

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?

Solution

An area of 100 cm² is equivalent to a surface of \(10\text{cm}\times 10\text{cm},\) which is much too small for the walls of a bedroom. An area of 100 km² is equivalent to a surface of \(10\text{km}\times 10\text{km},\) which is much too large for the walls of a bedroom. So, a reasonable value for the area of the walls is 100 m².

Condensed — the full section is in OpenStax Contemporary Mathematics.

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 .

Converting Units of Measure for Area Using Division

Try it.

A plot of land has an area of 237,500,000 m². What is the area in square kilometers?

Solution

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.

\[\frac{237,500,000}{1,000,000}=237.5\]

The plot of land has an area of 237.5 km².

Converting Units of Measure for Area Using Multiplication

Try it.

A plot of land has an area of 0.004046 km². What is the area of the land in square meters?

Solution

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.

\[0.004046\times 1,000,000=4,046\]

The plot of land has an area of 4,046 m².

Determining Area by Converting Units of Measure for Length First

Try it.

A computer chip measures 10 mm by 15 mm. How many square centimeters is the computer chip?

Solution

Step 1: Convert the measures of the computer chip into centimeters

\[\begin{array}{lll}10\text{mm} & = & 1\text{cm} \\ 15\text{mm} & = & 1.5\text{cm}\end{array}\]

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 \(1.5\) \({\text{cm}}^{\text{2}}\).

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.

Solving for the Area of Complex Surfaces

Try it.

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?

Solution

Step 1: Determine the area of the side of the building.

\[12m\times 9m=106{m}^{2}\]

Step 2: Determine the area of the door.

\[4m\times 3m=12{m}^{2}\]

Step 3: Subtract the area of the door from the area of the side of the building.

\[106{m}^{2}-12{m}^{2}=92{m}^{2}\]

So, you need to purchase \(92{m}^{2}\) of aluminum siding.

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.

Solving for Area with Distance Measurements of Different Units

Try it.

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?

Solution

Step 1: Use a conversion fraction to convert the information given in meters to kilometers.

\[1,250m\times \frac{1\text{km}}{1,000m}=1.25\text{km}\]

Step 2: Multiply to find the area.

\[2.2\text{km}\times 1.25\text{km}=2.75{\text{km}}^{2}\]

The park has an area of 2.75 km².

When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not relevant.

Solving for Area Using Multiple Steps

Try it.

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?

Solution

Step 1: Determine the area of the pantry floor in square centimeters.

\[100\text{cm}\times 200\text{cm}=20,000{\text{cm}}^{2}\]

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.

\[\frac{20,000}{10,000}=2\]

The area of the kitchen pantry floor is 2 m².

Step 3: Add the two areas of the pantry and the kitchen floors together.

\[15{m}^{2}+2{m}^{2}=17{m}^{2}\]

So, you need to buy 17 m² of tile.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two-dimensional space.
Area is measured in square meters units; in the metric system the base unit for area is square meters (\({\text{m}}^{2}\)).

Formulas

To determine the area of rectangular-shaped objects:

\(\text{Area}=\text{length }(l)\times \text{width }(w)\)

\[\begin{array}{l}\text{or} \\ A=l\times w\end{array}\]

You can convert between metric area units using the conversion factors shown in .

Practice (9)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Which unit of measure is most reasonable to describe the area of a sheet of paper: \({\text{km}}^{2}\), \({\text{cm}}^{2}\), or \({\text{mm}}^{2}\)?

Afslør svaret

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.
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?

Afslør svaret

An area of 100 cm² is equivalent to a surface of \(10\text{cm}\times 10\text{cm},\) which is much too small for the walls of a bedroom. An area of 100 km² is equivalent to a surface of \(10\text{km}\times 10\text{km},\) which is much too large for the walls of a bedroom. So, a reasonable value for the area of the walls is 100 m².
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.

Afslør svaret

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 plot of land has an area of 237,500,000 m². What is the area in square kilometers?

Afslør svaret

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.
\[\frac{237,500,000}{1,000,000}=237.5\]
The plot of land has an area of 237.5 km².
A plot of land has an area of 0.004046 km². What is the area of the land in square meters?

Afslør svaret

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.
\[0.004046\times 1,000,000=4,046\]
The plot of land has an area of 4,046 m².
A computer chip measures 10 mm by 15 mm. How many square centimeters is the computer chip?

Afslør svaret

Step 1: Convert the measures of the computer chip into centimeters
\[\begin{array}{lll}10\text{mm} & = & 1\text{cm} \\ 15\text{mm} & = & 1.5\text{cm}\end{array}\]
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 \(1.5\) \({\text{cm}}^{\text{2}}\).
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?

Afslør svaret

Step 1: Determine the area of the side of the building.
\[12m\times 9m=106{m}^{2}\]
Step 2: Determine the area of the door.
\[4m\times 3m=12{m}^{2}\]
Step 3: Subtract the area of the door from the area of the side of the building.
\[106{m}^{2}-12{m}^{2}=92{m}^{2}\]
So, you need to purchase \(92{m}^{2}\) of aluminum siding.
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?

Afslør svaret

Step 1: Use a conversion fraction to convert the information given in meters to kilometers.
\[1,250m\times \frac{1\text{km}}{1,000m}=1.25\text{km}\]
Step 2: Multiply to find the area.
\[2.2\text{km}\times 1.25\text{km}=2.75{\text{km}}^{2}\]
The park has an area of 2.75 km².
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?

Afslør svaret

Step 1: Determine the area of the pantry floor in square centimeters.
\[100\text{cm}\times 200\text{cm}=20,000{\text{cm}}^{2}\]
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.
\[\frac{20,000}{10,000}=2\]
The area of the kitchen pantry floor is 2 m².
Step 3: Add the two areas of the pantry and the kitchen floors together.
\[15{m}^{2}+2{m}^{2}=17{m}^{2}\]
So, you need to buy 17 m² of tile.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Measuring Area

Identify reasonable values for area applications.
Convert units of measures of area.
Solve application problems involving area.
area
square units
Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two-dimensional space.
Area is measured in square meters units; in the metric system the base unit for area is square meters (

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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