9.2 Measuring Area

A person painting a wall using a paint roller. — **Figure 9.5** A painter uses an extension roller to paint a wall.A painter uses an extension roller to paint a wall. (credit: "Paint Rollers are effective" by WILLPOWER STUDIOS/Flickr, CC BY 2.0)

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.

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.

$1 cm \times 1 cm = 1 {cm}^{2}$

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.

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

Units	Relationship	Conversion Rate
${km}^{2}$ to $m^{2}$	$\begin{matrix} km \times km & = & {km}^{2} \\ 1 km & = & 1,000 m \\ 1,000 m \times 1,000 m & = & 1,000,000 m^{2} \end{matrix}$	$1 ⁢ km squared equals 1,000,000 m squared$
$m^{2}$ to ${cm}^{2}$	$\begin{matrix} m \times m & = & m^{2} \\ 1 m & = & 100 cm \\ 100 cm \times 100 cm & = & 10,000 {cm}^{2} \end{matrix}$	$1 ⁢ m squared equals 10,000 ⁢ cm squared$
${cm}^{2}$ to ${mm}^{2}$	$\begin{matrix} cm \times cm & = & {cm}^{2} \\ 1 cm & = & 10 mm \\ 10 mm \times 10 mm & = & 100 {mm}^{2} \end{matrix}$	$1 cm squared equals 100 mm squared$

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.

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.

An illustration shows four units: millimeter squared, centimeter squared, meter squared, and kilometer squared. — **Figure 9.7** Common Conversion Factors for Metric Area Units

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.

Example 7 Solving for the Area of Complex Surfaces

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?

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

$12 m \times 9 m = 106 m^{2}$

Step 2: Determine the area of the door.

$4 m \times 3 m = 12 m^{2}$

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 $92 ⁢ m squared$ 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.

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

Example 9 Solving for Area Using Multiple Steps

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?

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

$100 cm \times 200 cm = 20,000 {cm}^{2}$

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.

Key Terms

area
square units

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 ( $m^{2}$ ).

Videos

Formulas

To determine the area of rectangular-shaped objects:

$Area = length (l) \times width (w)$

$\begin{matrix} or \\ A = l \times w \end{matrix}$

A rectangle with its longest side marked the length and the shortest side marked the width. — **Figure 9.8** Rectangle with Length $(l)$ and Width $(w)$ Labeled

You can convert between metric area units using the conversion factors shown in Figure 9.7.

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.