Opening the book…
Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For example, if you want to make authentic tamales, you might visit the nearest Hispanic grocery store. While shopping, you discover that there are two brands of mass for the same price, but one bag is marked \(1,200\) g and the other \(1\) kg. Which one is the better deal? Understanding the metric system allows you to understand that \(1,200\) grams (g) is equivalent to \(1.2\) kilogram (kg), so the \(1,200\) g bag is the better deal.
Units of Measurement in the Metric System
Units of measurement provide common standards so that, regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system:
- The metric unit for distance is the meter (m). A person’s height might be written as \(1.8\) meters (\(1.8\) m). A meter is slightly longer than a yard (\(3\) feet), while a centimeter is slightly less than half an inch.
- The metric unit for area is the square meter (m2). The area of a professional soccer field is \(7,140\) square meters (\(7,140\) \(m^2)\).
- The metric base unit for volume is the cubic meter (\(m^3\)). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in \(2\)-liter (\(2\) L) bottles.
- The gram (g) is a metric unit of mass, but is commonly used to express weight. The weight of a paper clip is approximately \(1\) gram (\(1\) g).
- The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be \(24\) °C.
While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system, a base unit is combined with a prefix, such as kilo in kilogram, to identify the relationship between smaller or larger units.
When using abbreviations for metric units, always separate the quantity and the units with a space, with no spaces between the letters or symbols in the units. For example, 7 millimeters is written as 7 mm, not 7mm.
Let’s be honest. Most of us use computers or smartphones to perform many of the calculations and conversions we were taught in math class. However, there is value in understanding the metric system, as it is ubiquitous, and, most importantly, knowing how metric units relate to each other allows you to compare prices, find the right tool in a workshop, or acclimate when in another country. No matter the circumstance, you cannot avoid the metric system.
While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature.
Converting Metric Units of Measure
Unlike the U.S. Customary System of Measurement, in which \(12\) inches is equal to \(1\) foot and \(3\) feet are equal to \(1\) yard, the metric system is structured so that the units within the system get larger or smaller by a power of \(10\).
The metric system combines base units and unit prefixes in a reasonable way, based on the size of the measured object or substance. The most used prefixes are shown in figure \(\PageIndex{2}\). An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic
King Henry Died By Drinking Chocolate Milk
Be careful about metric prefix name, its symbol, and its meaning.
For example, the kilo prefix has the symbol "k", its meaning is \(10^3\) or \(1000\), which means kilo is \(1000\) times larger than the base units.
Imagine you order a textbook online, and the shipping details indicate that the book weighs \(1\) kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is \(10^{3}\) or \(1,000\) times greater than a gram. One kilogram is equivalent to \(1,000\) grams.
The tip of a highlighter measures approximately \(1\) cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the of \(10^{-2}\) or \(\frac{1}{100}\) a meter. One meter is equivalent to \(100\) centimeters.
A conversion factor is used to convert between smaller metric units and larger metric units, and vice versa. It is a number that, when used in multiplication or division, converts units between metric systems that share the same base unit. In the metric system, these conversion factors are directly related to the powers of \(10\). The most commonly used conversion factors are shown in figure \(\PageIndex{3}\).
Because area is determined by multiplying two lengths, the area is the square of the length unit. Such as \(m^2\) or square meter. 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 chart.
Because volume is determined by multiplying three lengths, the volume is the cube of the length unit. Such as \(m^3\) or cubic meter. In other words, while a meter is \(100\) times greater in length than a centimeter, a cubic meter ( \(\mathrm{m}^3\) ) is \(100 \times 100\times 100\), or \(100, 000,000\) times greater in volume than a cubic centimeter \(\left(\mathrm{cm}^3\right)\). The relationships between benchmark metric volume units are shown in the following chart.
In figure \(\PageIndex{2}\), the units on the left are bigger than the units on the right.
- So we call \(1\) kg is \(1000\) times larger than 1 gram.
- Centimeter is \(10^{2}\) or \(100\) times smaller than a meter, while the kilometer is \(10^{3}\), or \(1,000\) times larger than a meter.
Example \(\PageIndex{3}\): Converting Metric Distances Using Multiplication
- The firehouse is \(13.45\) km from the library. How many meters is it from the firehouse to the library?
- How many liters is \(3,565\) milliliters?
- Convert \(125\) centigram to milligrams.
- Convert \(24,300,000\) milligrams to kilograms.
- Change \(0.025\) kilograms to centigram
- Answer
-
- When converting from a larger unit to a smaller unit, use multiplication, see in the figure \(\PageIndex{2}\). We need to multiply by \(10\) three times. It is the same as saying multiply by \(1000\) or move the decimal \(3\) places to the right. \[13.45\times 1,000=13,450 \nonumber\] So, the firehouse is \(13,450\) meters away from the firehouse.
- When converting from a smaller unit to a larger unit, use division, see in the figure \(\PageIndex{2}\). We need to divide by \(10\) three times. It is the same as saying divide by \(1,000\) or move the decimal \(3\) places to the left. \[3,565\div 1,000=3.565 \nonumber\] So, \(3,565\) milliliters is \(3.565\) liters.
- When converting from a larger unit to a smaller unit, use multiplication to calculate the result. See in the figure \(\PageIndex{2}\). We need to multiply by \(10\). It is the same as saying move the decimal \(1\) places to the right.. \[125\times10= 1,250\nonumber\] So, \(125\) centigrams are \(1,250\) milligrams.
- Use division to convert from a smaller metric weight unit to a larger metric weight unit. To convert from milligrams to kilograms, divide the value of the weight in milligrams by \(1,000\) to first convert from milligrams to grams and divide by \(1,000\) again to convert from grams to kilograms.
\[\frac{24,300,000}{1,000,000}=24.3 \nonumber\]
So, \(24,300,000\) milligrams are equivalent to \(24.3\) kilograms.
- Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms to centigrams, multiply the value of the weight by \(10\), five times .\[0.025\times 10\times10\times10\times10\times10=2,500\nonumber\] So the \(0.025\) kilogram is equivalent to \(2,500\) centigrams.
Example \(\PageIndex{4}\): Identifying Reasonable Units for Weight
- Which is the more reasonable value for the weight of a newborn baby: \(3.5\) kg or \(3.5\) g?
- Which of the following represents a reasonable value for the weight of three lemons? \(250\) g, \(2,500\)g, or \(250\) kg?
- What is a reasonable value for the length of a person’s thumb: \(5\) meters, \(5\) centimeters, or \(5\) millimeters?
- What is a reasonable value for the width of an electric chord: \(4\) meters, \(4\) centimeters, or \(4\) millimeters?
- The blue whale is the largest living mammal on Earth. Which of the following is a reasonable value for the weight of a blue whale: \(149\) g, \(149\) kg, or \(149\) mt? Explain your answer.
- Answer
-
- Using our reference weights, a baby weighs more than \(3.5\) paperclips, so \(3.5\) kilograms is a more reasonable value for the weight of a newborn baby.
- Because a kilogram is about \(2.2\) pounds, we can eliminate \(250\) kg as it is way too heavy. \(2,500\) grams is equivalent to \(2.5\) kilograms, or about five pounds, which is again, too heavy. So, a reasonable value for the weight of three lemons would be \(250\) grams.
- Given that a meter is slightly longer than a yard, \(5\) meters is not a reasonable value for the length of a person’s thumb. Since a millimeter is \(10\) times smaller than a centimeter, which is approximately half an inch, \(5\) millimeters is not a reasonable estimate for the length of a person’s thumb. The correct answer is \(5\) centimeters.
- \(4\) millimeters.
- A reasonable value for the weight of a blue whale is \(149\) metric tons. Both \(149\) g and \(149\) kg are much too small a value for the largest living mammal on Earth.
It is impossible to weigh a living whale. Fredrik Christiansen from the Aarhus Institute of Advanced Studies in Denmark developed an innovative way to measure the weight of whales. Using images taken from a drone and computer modeling, the weight of a whale can be estimated with great accuracy
Did you know that the metric system pervades daily life in the United States already? While Americans still may purchase gallons of milk and measure house sizes in square feet, there are many instances of the metric system. Photographers buy 35 mm film and use 50 mm lenses. When you have a headache, you might take 600 mg of ibuprofen. And if you are eating a low-carb diet you probably restrict your carb intake to fewer than 20 g of carbs daily. Did you know even the dollar is metric? In the video, Neil DeGrasse Tyson and comedian co-host Chuck Nice provide an amusing perspective on the metric system.
The International System of Units (SI) is the current international standard metric system and is the most widely used system around the world. In most English-speaking countries, SI units such as meter, liter, and metric ton are spelled meter, liter, and tonne.
Just how much is the metric system a part of your life now? Probably more than you think. For the next \(24\) hours, take notice as you move through your daily activities. When you are shopping, are the package sizes provided in metric units? Change the weather app on your phone to display the temperature in degrees Celsius. Are you able to tell what kind of day it will be now? While the United States is not officially using the metric system, you will still find the metric system all around you
Measuring Areas
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.
\[1 \mathrm{~cm} \times 1 \mathrm{~cm} =1\mathrm{~cm}^2 \nonumber \]
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 the 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 \(\PageIndex{3}\). Given this limitation, the basic formula for area is:
\[\text{Area}=\text{lenght}\times\text{width}\nonumber\]
A fundamental understanding of metric area is essential for identifying 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.
Example \(\PageIndex{5}\): Determining Reasonable Units for Area
- Which unit of measure is most reasonable to describe the area of a sheet of paper: \(\mathrm{km}^2, \mathrm{~cm}^2\), or \(\mathrm{mm}^2\) ?
- You want to paint the walls of your bedroom. Which represents a reasonable value for the area of the walls: \(100\) cm2, \(100\) m2, or \(100\) km2
- Answer
-
- In the U.S. Customary System of Measurement, the length and width of paper are 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.
- 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 of \(10\) 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.
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 \(\PageIndex{5}\).
Example \(\PageIndex{6}\): Converting Units of Measure for Area
- A plot of land has an area of \(237,500,000\) m2. What is the area in square kilometers?
- A plot of land has an area of \(0.004046\) km2. What is the area of the land in square meters?
- Answer
-
- Use division to convert from a smaller metric area unit to a larger metric area unit. To convert from \( \mathrm{~m}^2\) to \( \mathrm{~km}^2\), divide the value of the area by \(1,000,000\) (divide the value of the area of by \(100\) three times). See figure \(\PageIndex{5}\) or move the decimal six units to the left.\[\frac{237,500,000}{1,000,000}=237.5\nonumber\] So the area of the plot is \(237.5 \mathrm{~km}^2\)
- Use multiplication to convert from a larger metric area unit to a smaller metric area unit. To convert from \( \mathrm{~km}^2\) to \( \mathrm{~m}^2\), multiply the value of the area by \(1,000,000\) or move the decimal six decimal places to the right. \[0.004046\times 1,000,000=4,046\nonumber\] So the area of the plot is \(4,406 \mathrm{~m}^2\)
Your Turn \(\PageIndex{7}\): Determining Area by Converting Units of Measure for Length First
- A computer chip measures \(10\) mm by \(15\) mm. How many square centimeters is the computer chip?
- A piece of fabric measures \(110\) cm by \(116\) cm. What is the area of the fabric in square meters?
- Answer
-
1. First, convert the measures of the computer chip into centimeters
\[\begin{array}{l}
10 \mathrm{~mm}=1 \mathrm{~cm} \\
15 \mathrm{~mm}=1.5 \mathrm{~cm}
\end{array} \nonumber \]Use the area formula to determine the area of the chip.
\[1 \times 1.5=1.5 \nonumber \]
The computer chip has an area of \(1.5 \mathrm{~cm}^2\).
2. First, convert the measures of the fabric into meters from centimeters.
Use the area formula to determine the area of the fabric
\[1.10\mathrm{~m} \times 1.16 \mathrm{~m}=1.276 \mathrm{~m}^2 \nonumber \]
So the area of the fabric is \(1.276 \mathrm{~m}^2\).
When calculating area, ensure that both distance measurements are expressed in the same 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 irrelevant areas.
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.
A hectare is a metric unit used to measure area, specifically for large plots of land like farms, forests, and parks. Imagine a perfect square where each side is \(100\) meters long (about \(328\) feet). The space inside that square is \(1\) hectare.
Measuring Volume
Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the SUV can hold.
Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine the volume in square units:
\[1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}=1 \mathrm{~cm}^3 \nonumber \]
Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, to find a volume in cubic centimeters, each of the measures must be in centimeters.
The formula used to determine volume depends on the shape of the three-dimensional object. Here we will limit our discussions to the area to rectangular prisms like the one in Figure Given this limitation, the basic formula for volume is:
\[V=l\times w\times h\nonumber\]
To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given volume is reasonable for the situation.
Example \(\PageIndex{8}\): Determining Reasonable Values for Volume
- A grandparent wants to send cookies to their grandchild away at college. Which represents a reasonable value for the volume of a box to ship the cookies: \(3,375\) km3, \(3,375\) m3, or \(3,375\) cm3?
- A food manufacturer is prototyping new packaging for one of its most popular products. Which represents a reasonable value for the volume of the box:
\(2\) dm3, \(2\) cm3, or \(2\) mm3 ?
- Answer
-
- A volume of \(3,375\) \(\mathrm{~km}^3\) is equivalent to a rectangular prism with dimensions of \(15\mathrm{~km}\times15\mathrm{~km}\times15 \mathrm{~km}\), which is far too large for a shipping box. An area of \(3,375\) \(\mathrm{~m}^3\) is equivalent to a surface of \(15\mathrm{~m}\times15\mathrm{~m}\times15 \mathrm{~m}\), which is also too large. A reasonable value for the volume of the box is \(3,375\) \(\mathrm{~cm}^3\).
- A decimeter is equal to \(10\) centimeters. A box with a volume of \(2\) \(\mathrm{~dm}^3\)might have the dimensions \(1 \mathrm{~dm}\times1 \mathrm{~dm}\times1 \mathrm{~dm}\), or \(10\mathrm{~cm}\times10 \mathrm{~cm}\times10 \mathrm{~cm}\) , which is reasonable. A box with a volume of \(2\) \(\mathrm{~cm}^3\) or \(2\) \(\mathrm{~mm}^3\) would be too small.
Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in Figure \(\PageIndex{3}\).
Example \(\PageIndex{9}\): Converting Units of Measure for Volume
- A pencil case has a volume of \(1,700\) cm3. What is the volume in cubic millimeters?
- A shipping container has a volume of \(33.2\) m3. What is the volume in cubic centimeters?
- A holding tank at the local aquarium has a volume of \(22,712,000,000\) \(\mathrm{~dm}^3\). What is the volume in cubic meters?
- Answer
-
- Use multiplication to convert from a larger metric volume unit to a smaller metric volume unit. To convert from \(\mathrm{~cm}^3\) to \(\mathrm{~mm}^3\), multiply the value of the volume by \(1,000\). \[1,700\times1,000=1,700,000\nonumber\] The pencil case has a volume of \(1,700,000\) \(\mathrm{~mm}^3\).
- Use multiplication to convert a larger metric volume unit to a smaller metric volume unit. To convert from \(\mathrm{~m}^3\)to \(\mathrm{~cm}^3\), first multiply the value of the volme by \(1,000\) to convert from \(\mathrm{~mm}^3\) to \(\mathrm{~dm}^3\) and then multiply again by \(1,000\) to convert form \(\mathrm{~dm}^3\) to \(\mathrm{~m}^3\). \[33.2\times1,000\times1,000=33,200,000\nonumber\]The shipping container has a volume of \(32,200,000\) \(\mathrm{~cm}^3\).
- Figure \(\PageIndex{3}\) indicates that when converting from a smaller metric volume unit to a larger metric volume unit you divide using the given conversion factor. To convert from \(\mathrm{~cm}^3\) to \(\mathrm{~m}^3\), divide the value of the volume by \(1,000\) to first convert from \(\mathrm{~cm}^3\) to\(\mathrm{~dm}^3\), then divide again to convert from \(\mathrm{~dm}^3\) to \(\mathrm{~m}^3\). Or divide by \(1,000,000\). \[\frac{22,712,000,000}{1,000,000}=22,712\nonumber\] The holding tank has a volume of \(22,712\) \(\mathrm{~m}^3\).
Understanding Other Metric Units of Volume
When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to \(1\) cubic decimeter. A milliliter is equal in volume to \(1\) cubic centimeter. So, when a doctor orders \(10\)cc (cubic centimeters) of saline to be administered to a patient, they are referring to \(10\) \(\mathrm{mL}\) of saline.
The most frequently used factors for converting from cubic meters to liters are listed in the following table.
| Unit to Convert | Conversation Factor |
| \(\mathrm{~cm}^3\; to \;\mathrm{mL}\) |
\(1 \mathrm{~cm}^3\; = \;1\mathrm{mL}\) OR \(1,000 \mathrm{~cm}^3\; = \;1\mathrm{L}\) |
| \(\mathrm{~dm}^3\; to \;\mathrm{L}\) | \(1 \mathrm{~dm}^3\; = \;1\mathrm{L}\) |
| \( \mathrm{~m}^3\; to \;\mathrm{kL}\) | \(1\mathrm{~m}^3\; = \;1\mathrm{kL}\) |
Example \(\PageIndex{10}\): Converting Different Units of Measure for Volume
A holding tank at the local aquarium has a volume of \(22,712,000,000 \mathrm{~cm}^3\)? What is the capacity of the holding tank in liters?
- Answer
-
Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from \(\mathrm{~cm}^3\) to \(\mathrm{L}\) by dividing the value of the volume in \(\mathrm{~cm}^3\) by \(1,000\).
\[ \begin{aligned} 22,712,000,000 \text{ cm\(^3\)}&= 22,712,000,000 \text{ cm\(^3\) } \times \frac{1\text{ L}}{1,000 \text{ cm\(^3\)}} \\
&=22,712,000,000~ \cancel{\text{cm\(^3\)}} \times \frac{1 \text{ L}}{1,000 ~ \cancel{\text{cm\(^3\)}}}\
& \textcolor{red}{ \text{ Cancel common units.}} \\
&= 22,712,000 \text{ L }
\end{aligned}\nonumber \]The holding tank holds \(22,712,000\) \(\mathrm{L}\) of water.
3.2: The Metric System is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.
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