6.3.1: Temperature Scales
- Page ID
- 62188
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- Convert from one temperature scale to the other, using conversion formulas.
Introduction
Turn on the television any morning and you will see meteorologists talking about the day’s weather forecast. In addition to telling you what the weather conditions will be like (sunny, cloudy, rainy, muggy), they also tell you the day’s forecast for high and low temperatures. A hot summer day may reach 100o in Philadelphia, while a cool spring day may have a low of 40o in Seattle.
If you have been to other countries, though, you may notice that meteorologists measure heat and cold differently outside of the United States. For example, a TV weatherman in San Diego may forecast a high of 89o, but a similar forecaster in Tijuana, Mexico, which is only 20 miles south, may look at the same weather pattern and say that the day’s high temperature is going to be 32o. What’s going on here?
The difference is that the two countries use different temperature scales. In the United States, temperatures are usually measured using the Fahrenheit scale, while most countries that use the metric system use the Celsius scale to record temperatures. Learning about the different scales, including how to convert between them, will help you figure out what the weather is going to be like, no matter which country you find yourself in.
Measuring Temperature on Two Scales
Fahrenheit and Celsius are two different scales for measuring temperature.
| A thermometer measuring a temperature of 22o Celsius is shown here. |  |
A thermometer measuring a temperature of 72o Fahrenheit shown here. |
| On the Celsius scale, water freezes at 0o and boils at 100o. | On the Fahrenheit scale, water freezes at 32o and boils at 212o. | |
| If the United States were to adopt the Celsius scale, forecast temperatures would rarely go below -30o or above 45o. (A temperature of -18o may be forecast for a cold winter day in Michigan, while a temperature of 43o may be predicted for a hot summer day in Arizona.) | In the United States, forecast temperatures measured in Fahrenheit rarely go below -20o or above 120o. (A temperature of 0o may be forecast for a cold winter day in Michigan, while a temperature of 110o may be predicted for a hot summer day in Arizona.) | |
| Most office buildings maintain an indoor temperature between 18o Celsius and 24o Celsius to keep employees comfortable. | Most office buildings maintain an indoor temperature between 65o Fahrenheit and 75o Fahrenheit to keep employees comfortable. |
A cook puts a thermometer into a pot of water to see how hot it is. The thermometer reads 132o, but the water is not boiling yet. Which temperature scale is the thermometer measuring?
- Celsius
- Fahrenheit
- Answer
-
- Incorrect. On the Celsius scale, water boils at 100o, so if the water is not boiling and the measurement is over 100o, then it cannot be Celsius. The correct answer is Fahrenheit.
- Correct. Water boils at 212o on the Fahrenheit scale, so a measurement of 132o on a Fahrenheit scale is legitimate for hot (but non-boiling) water.
Converting Between the Scales
By looking at the two thermometers shown, you can make some general comparisons between the scales. For example, many people tend to be comfortable in outdoor temperatures between 50o Fahrenheit and 80o Fahrenheit (or between 10o Celsius and 25o Celsius). If a meteorologist predicts an average temperature of 0o Celsius (or 32o Fahrenheit), then it is a safe bet that you will need a winter jacket.
Sometimes, it is necessary to convert a Celsius measurement to its exact Fahrenheit measurement or vice versa. For example, what if you want to know the temperature of your child in Fahrenheit, and the only thermometer you have measures temperature in Celsius measurement? Converting temperature between the systems is a straightforward process as long as you use the formulas provided below.
To convert a Fahrenheit measurement to a Celsius measurement, use this formula.
\(\ C=\frac{5}{9}(F-32)\)
To convert a Celsius measurement to a Fahrenheit measurement, use this formula.
\(\ F=\frac{9}{5} C+32\)
How were these formulas developed? They came from comparing the two scales. Since the freezing point is 0o on the Celsius scale and 32o on the Fahrenheit scale, we subtract 32 when converting from Fahrenheit to Celsius, and add 32 when converting from Celsius to Fahrenheit.
There is also a reason for the fractions \(\ \frac{5}{9}\) and \(\ \frac{9}{5}\). There are 100 degrees between the freezing (0o) and boiling points (100o) of water on the Celsius scale and 180 degrees between the similar points (32o and 212o) on the Fahrenheit scale. Writing these two scales as a ratio, \(\ \frac{F^{\circ}}{C^{\circ}}\), gives \(\ \frac{180^{\circ}}{100^{\circ}}=\frac{180^{\circ} \div 20}{100^{\circ} \div 20}=\frac{9}{5}\). If you flip the ratio to be \(\ \frac{C^{\circ}}{F^{\circ}}\), you get \(\ \frac{100^{\circ}}{180^{\circ}}=\frac{100^{\circ} \div 20}{180^{\circ} \div 20}=\frac{5}{9}\). Notice how these fractions are used in the conversion formulas.
The example below illustrates the conversion of Celsius temperature to Fahrenheit temperature, using the boiling point of water, which is 100o Celsius.
The boiling point of water is 100o Celsius. What temperature does water boil at in the Fahrenheit scale?
Solution
| \(\ F=\frac{9}{5} C+32\) | A Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula at the left. |
|
\(\ F=\frac{9}{5}(100)+32\) \(\ F=\frac{900}{5}+32\) |
Substitute 100 for \(\ C\) and multiply. |
|
\(\ F=\frac{900 \div 5}{5 \div 5}+32\) \(\ F=\frac{180}{1}+32\) |
Simplify \(\ \frac{900}{5}\) by dividing numerator and denominator by 5. |
| \(\ F=212\) | Add 180+32 |
The boiling point of water is 212o Fahrenheit.
Water freezes at 32o Fahrenheit. On the Celsius scale, what temperature is this?
Solution
| \(\ C=\frac{5}{9}(F-32)\) | A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left. |
| \(\ C=\frac{5}{9}(32-32)\) | Substitute 32 for \(\ F\) and subtract. |
| \(\ C=\frac{5}{9}(0)\) | Any number multiplied by 0 is 0. |
| \(\ C=0\) |
The freezing point of water is 0o Celsius.
The two previous problems used the conversion formulas to verify some temperature conversions that were discussed earlier: the boiling and freezing points of water. The next example shows how these formulas can be used to solve a real-world problem using different temperature scales.
Two scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of 120o Celsius; the other gets a result of 250o Fahrenheit. Which temperature is higher and by how much?
Solution
| What is the difference between 120o Celsius and 250o Fahrenheit? | One temperature is given in degrees Celsius, and the other is given in degrees Fahrenheit. To find the difference between them, we need to measure them on the same scale. |
| \(\ F=\frac{9}{5} C+32\) |
Use the conversion formula to convert 120o Celsius to degrees Fahrenheit. (You could convert 250o Fahrenheit to degrees Celsius instead; this is explained in the text after this example.) |
| \(\ F=\frac{9}{5}(120)+32\) | Substitute 120 for \(\ C\). |
| \(\ F=\frac{1080}{5}+32\) | Multiply. |
| \(\ F=\frac{1080 \div 5}{5 \div 5}+32\) | Simplify \(\ \frac{1080}{5}\) by dividing numerator and denominator by 5. |
| \(\ F=\frac{216}{1}+32\) | Add \(\ 216+32\). |
| \(\ F=248\) | You have found that \(\ 120^{\circ} \mathrm{C}=248^{\circ} \mathrm{F}\). |
| \(\ 250^{\circ}-248^{\circ}=2^{\circ} \text { Fahrenheit }\) | To find the difference between 248o Fahrenheit and 250o Fahrenheit, subtract. |
250o Fahrenheit is the higher temperature by 2o Fahrenheit.
You could have converted 250o Fahrenheit to degrees Celsius instead, and then found the difference in the two measurements. (Had you done it this way, you would have found that 250oF=121.1oC, and that 121.1o Celsius is 1.1o Celsius higher than 120o Celsius.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.
Tatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around 26o Celsius. What is the average temperature in degrees Fahrenheit?
- \(\ 79^{\circ} \text { Fahrenheit }\)
- \(\ -3^{\circ} \text { Fahrenheit }\)
- \(\ 45^{\circ} \text { Fahrenheit }\)
- \(\ 58^{\circ} \text { Fahrenheit }\)
- Answer
-
- Correct. Tatiana can find the Fahrenheit equivalent by solving the equation \(\ F=\frac{9}{5}(26)+32\). The result is 78.8o Fahrenheit, which rounds to 79o Fahrenheit.
- Incorrect. You used the wrong formula. To find the Fahrenheit equivalent, use the formula \(\ F=\frac{9}{5} C+32\). The correct answer is 79o Fahrenheit.
- Incorrect. You misapplied the formula; try substituting 26 for \(\ C\) in the formula \(\ F=\frac{9}{5} C+32\). The correct answer is 79o Fahrenheit.
- Incorrect. You misapplied the formula; try substituting 26 for \(\ C\) in the formula \(\ F=\frac{9}{5} C+32\). The correct answer is 79o Fahrenheit.
Summary
Temperature is often measured in one of two scales: the Celsius scale and the Fahrenheit scale. A Celsius thermometer will measure the boiling point of water at 100o and its freezing point at 0o; a Fahrenheit thermometer will measure the same events at 212o for the boiling point of water and 32o as its freezing point. You can use conversion formulas to convert a measurement made in one scale to the other scale.