# Temperature Conversion Formula - Celsius to Fahrenheit Equation

## Temperature Conversion

The most common units of temperature measurement are **degrees Celsius **, which is denoted as °C, **degrees Fahrenheit**, denoted as °F, and Kelvin, denoted as K. Perhaps the reader is more acquainted with the former two units, given the fact that they are widely used in everyday applications. Degrees Fahrenheit is used in the United States and other countries, a few of which are Bahamas, Palau, and Belize. Whereas the degrees Celsius is used by countries that adopted the metric system, mainly countries in Europe, Asia, and some countries in America, including Canada. The Kelvin unit is used primarily in scientific calculations and research related to temperature, such as problems that involve the ideal gas law.

In this lesson, the relation between each temperature unit will be discussed thoroughly, and the unit temperature conversion formula for each will be derived. The purpose of converting from one temperature unit to another is to express the same temperature in a unit one is more familiar with. Let's suppose that Alice is more familiar with temperatures expressed in degrees Celsius and that she is planning to go on a trip to the United States of America, where the temperature is expressed in Fahrenheit. Upon planning for the trip, she first examines the weather conditions in the region she will be visiting, and she learns that this week the weather will be pleasant, the skies will be clear, and that it's a perfect time for a holiday, except that it's 68 degrees. At first glance, Alice would think twice before going on this trip since the temperature is a staggering 68 degrees, well, in Fahrenheit. Should she convert this temperature to Celsius, she'd confirm that the weather is indeed pleasant and that it's a perfect time for a holiday.

## Solving Problems With Temperature

Planning a trip to Vancouver from Boston includes gathering lots of information. It helps to know local temperatures while deciding on the appropriate clothing to bring. Temperatures in Canada are in **degrees Celsius** while in the United States, the temperature is measured in **degrees Fahrenheit**.

With Internet resources, these conversions are easily done but have you ever wondered how to do them yourself? With one simple equation, the conversions with Celsius and Fahrenheit temperatures are easily made. Have you ever wondered where this equation came from? Then there's the temperature measure called the Kelvin. In this lesson, we will see how they are all related.

## Easy Temperature Conversion Formulas

The following table shows the temperatures in Celsius and in Fahrenheit. As the reader can see, 15°C is equal to 59°F, and that 113°F is equal to 45°C.

Celsius | Fahrenheit |
---|---|

-55 | -67 |

-50 | -58 |

-45 | -49 |

-40 | -40 |

-35 | -31 |

-30 | -22 |

-25 | -13 |

-20 | -4 |

-15 | 5 |

-10 | 14 |

-5 | 23 |

0 | 32 |

5 | 41 |

10 | 50 |

15 | 59 |

20 | 68 |

25 | 77 |

30 | 86 |

35 | 95 |

40 | 104 |

45 | 113 |

50 | 122 |

By examining this table alone, it would be difficult to deduce the relationship between these two units. To understand this relationship, we will start by discussing the thermodynamic properties of water (as a basis) then use the standard line equation to derive the conversion formulas.

The following image shows the freezing and the boiling temperatures of water in both Celsius and Fahrenheit. At atmospheric pressure, the water boils at 100°C and it freezes at 0°C. The temperature difference between these two points is equal to {eq}\Delta C = 100 - 0 = 100 {/eq}. And in the Fahrenheit scale, water boils at 212°F and freezes at 32°F, the difference between the temperatures is {eq}\Delta F = 212 - 32 = 180 {/eq}.

Recall the line equation, y = mx + b, where m is the slope of the line and b is the constant. Let us first find the slope m, which would be equal to the ratio of the difference in temperatures in Fahrenheit to the difference in Celsius, as shown: {eq}m = \Delta F / \Delta C = 180/100 = 1.8 = 9/5 {/eq}. Next, we will rewrite the line equation this way {eq}T (F) = m T (C) + b {/eq}, which is {eq}T (F) = 1.8 T (C) + b {/eq}

The only component that we have yet to identify is the constant b. Recall that the freezing temperatures of water in Celsius and in Fahrenheit are 0 and 32 degrees respectively. To solve for the constant b in the line equation, we will need a slope and a point. We have already found the slope, and the point that we will be using is the freezing point of water. Substituting the values in the previous equation gives:

{eq}32 (F) = 1.8 * 0 (C) + b {/eq}

Solving for b:

{eq}b = 32 {/eq}

We have identified the slope and the constant for the line equation. Ultimately, we have derived the Celsius to Fahrenheit conversion formula. The following equation is the conversion formula one must use when converting a temperature from Celsius to Fahrenheit.

{eq}T (F) = 1.8 T (C) + 32 {/eq}

To find the Fahrenheit to Celsius conversion formula, one can simply mathematically manipulate the equation discussed earlier, the final format would be the following:

{eq}T (C) = (T (F) - 32)/1.8 {/eq}

To summarize:

**Convert C to F:** {eq}T (F) = 1.8 T (C) + 32 {/eq}

**Convert F to C:** {eq}T (C) = (T (F) - 32)/1.8 {/eq}

### Other Temperature Conversion Formulas

There are other formulas that can be used to convert temperatures in Celsius and Fahrenheit to Kelvin, which is a temperature unit used to express absolute temperatures, its null point is the absolute zero, which is the lowest temperature possible. The reason why it is important to know how to convert to Kelvin is because this unit is widely used in solving scientific and engineering problems.

**Convert C to K:** {eq}T ( K ) = T (C) + 273 {/eq}

**Convert K to C:** {eq}T ( C ) = T (K) - 273 {/eq}

**Convert F to K:** {eq}T ( K ) = ( T (F) - 32 )/1.8 + 273 {/eq}

**Convert K to F:** {eq}T (F) = ( T (K) - 273) * 1.8 + 32 {/eq}

### Celsius, Fahrenheit & Kelvin Equation Examples

- Example 1: The weather reporter said that the temperature is 100 degrees Fahrenheit and that the ambience is humid. Convert this temperature to Celsius.

{eq}T (C) = (T (F) - 32)/1.8 = (100 - 32)/1.8= 37.7C {/eq}

- Example 2: Your weather app reported the following: the weather is mostly cloudy in Istanbul, the humidity is at 67%, and the temperature is 295K. Convert the temperature to Celsius.

{eq}T(C) = T(K) + 273 = 295 - 273 = 22 C {/eq}

- Example 3: You are in a middle of a thermodynamics test and you were asked to find the temperature of an object then to draw conclusions using a provided graph. You found the temperature successfully, and it turned out to be equal to 45°C. The temperature values in the graph is in Kelvin, convert to Kelvin to draw conclusions.

{eq}T (K) = T(C) + 273 = 45 + 273 = 318K {/eq}

- Example 4: You are still solving the thermodynamics test and you were asked to solve a problem using the ideal gas law. You recall that the temperature must always be absolute when applying this law, the temperature is given as 88°F. Convert to Kelvin.

{eq}T (K) = ( T (F) - 32 )/1.8 + 273 = (88 - 32)/1.8 + 273 = 304 K {/eq}

- Example 5: Your scientist friend announced that tonight's weather is clear and that the temperature is 300K. Convert to Fahrenheit before everyone panics.

{eq}T (F) = ( T (K) - 273) * 1.8 + 32 = (300 - 273)*1.8 + 32 = 80.6 F {/eq}

- Example 6: You are discussing going on a trip to the Grand Canyon with your friends, one of your friends opposed the idea and said that the temperature is 20 degrees (they deliberately failed to mention Celsius but you saw past that), and that you will all freeze to death if you went there. Convert this temperature to Fahrenheit and save your holiday trip.

{eq}T (F) = 1.8 * T (C) + 32 = 1.8 * 20 + 32 = 68F {/eq}

## Celsius to Fahrenheit Ratio

Should the reader examine the graph image in the 'Easy Temperature Conversion Formulas' section, they would deduce that the relationship between the temperature in Celsius and in Fahrenheit is directly proportional, and that the slope of the line is equal to 9/5, which is simply the ratio we have been using, 1.8. Now, is there any point where the temperature in Celsius is exactly equal to the temperature in Fahrenheit? To answer this question, we need to set the conversion factor for both temperature scales and make them equal to each other, then we will solve for the temperature.

{eq}T (F) = 1.8 * T (C) + 32 {/eq}

{eq}x = 1.8 * x + 32 {/eq}

{eq}x - 1.8 * x = 32 {/eq}

{eq}- 0.8 x = 32 {/eq}

{eq}x = 32/(-0.8) {/eq}

{eq}x = - 40 {/eq}

This shows that the Celsius and the Fahrenheit scales have one point at which they intersect, this point is called the **crossing point**. This is illustrated by the following graph. Put simply, T = -40°C = -40°F.

Another temperature scale that was briefly discussed was the Kelvin scale. Unlike the Fahrenheit and the Celsius scales, the temperature in Kelvin is always absolute, meaning it is never negative. And that is because the first point on the Kelvin scale is the zero, the **absolute zero**. The absolute zero is the lowest temperature possible at which a system is in the state of the lowest possible energy.

To summarize:

**Absolute zero value in each scale:** 0K = -273 °C = - 459.7°F

**Boiling point of water in each scale:** 373K = 100 °C = 212°F

**Freezing point of water in each scale:** 273K = 0 °C = - 459.7°F

**The crossing point:** -40 °C = - 40°F

## Lesson Summary

**Temperature conversion formulas** can be derived by drawing the relation between each temperature scale and using the standard line equation. Temperatures are traditionally expressed either in **Celsius**, in most countries that have adopted the metric system, or in **Fahrenheit**, mainly in the United States and a few other countries. One degree Celsius is larger than one degree Fahrenheit. To convert between each unit, we use the freezing and boiling points of water in both scales, find the ratio/slope, then use the freezing point in each scale to find the constant b of the line equation, which results in the following formula: {eq}T (F) = 1.8 T (C) + 32 {/eq}.

**Kelvin** is often used in scientific and engineering applications. It is an absolute temperature scale whose null point is **absolute zero**. Absolute zero is the lowest possible temperature that a system can reach. It is represented as 0K and it corresponds to -273°C and -459.7°F. The value of the **crossing point** is -40. The crossing point is a point where the Celsius and Fahrenheit scales intersect, meaning that at -40°C the corresponding temperature in Fahrenheit is -40°F.

## The Celsius-Fahrenheit Equation

The temperature in Boston this fine summer morning is 72o F while at the same time in Vancouver, the people are enjoying 15o C. To get to the same temperature scale, whether it's degrees Celsius or degrees Fahrenheit, means that these measurements have a relationship. We can convert from one to the other.

At sea level, water boils at 212o F and freezes at 32o F. That's a difference in temperatures of 212 - 32 = 180. You could say Î”F = 180. On the Celsius scale, water boils at 100o C and freezes at 0o C; a temperature difference Î”C = 100.

Let's think about a slope, *m* = Î”F / Î”C. The slope is Î”F / Î”C = 180/100 = 1.8.

Remember the equation of a line expressed as *y* = *mx* + *b* ? Well, what if we had *F* = *mC* + *b*? So, *F* = 1.8*C* + *b*.

What about the constant *b*? When water freezes at *C* = 0 we have *F* = 32. In our equation, *F* = 1.8*C* + *b*, let *C* = 0 and *F* = 32 to get 32 = 1.8(0) + *b*. Meaning, 32 = 0 + *b* or *b* = 32.

We're done! The beautiful conversion equation between *C* and *F* is:

*F*= 1.8*C*+ 32.

### Example 1

Let's look at an example. Our friends in Boston would like to know what a Vancouver temperature of 15o C is in degrees Fahrenheit.

From *F* = 1.8*C* + 32 let *C* = 15. Then, *F* = 1.8(15) + 32 = 59. Therefore, in Boston we have 72o F while in Vancouver it's 59o F. Big difference! 72 - 59 = 13 degrees.

### Example 2

Here's another example: our friends in Vancouver would like to know what the Boston temperature of 72o F is in Celsius.

From *F* = 1.8*C* + 32 , solve for *C* by subtracting 32 from both sides:

*F*- 32 = 1.8*C*+ 32 - 32; which simplifies to*F*- 32 = 1.8*C*. Then divide both sides by 1.8- (
*F*- 32) / 1.8 = 1.8*C*/ 1.8; which simplifies to - (
*F*- 32) / 1.8 =*C*

So, *C* = (*F* - 32) / 1.8 = (72 - 32) / 1.8 â‰… 22.

Boston is at 22o C while Vancouver is at 15o C. Only a 7o C change. The difference in temperature sounds better in degrees Celsius.

### Example 3

How about calculating the temperature in o F for another place in North America? How about Death Valley, CA,? Now here's a third example: the measured temperature in Death Valley is currently 35o C. What is this temperature in degrees Fahrenheit?

In degrees Fahrenheit this is *F* = 1.8*C* + 32 = 1.8(35) + 32 = 95.

Not bad, 95o F. However, on July 10, 1913, the hottest temperature ever recorded in the world was in Death Valley. This temperature was 134o F.

### Example 4

Here's one more example: our friends in Canada can't relate to this 134o F. Let's convert to degrees Celsius.

*C* = (*F* - 32) / 1.8 = (134 - 32) / 1.8 â‰… 57o C. More than half way up the Celsius scale from freezing towards boiling. No wonder it's called Death Valley!

## The Crossing Point

Is there a temperature value where *C* and *F* are the same? Turns out there is, and this value is -40. So if the thermometer reads -40o C in Vancouver, and it reads -40o F in Boston, the temperatures are the same.

To show this, we take the *F* = 1.8*C* + 32 equation and replace *F* with *C*. Then:

*C*= 1.8*C*+ 32. Subtract 1.8*C*from both sides:*C*- 1.8*C*= 1.8*C*+ 32 - 1.8*C*; then you simplify- -0.8
*C*= 32; divide both sides by -0.8: - -0.8
*C*/ (-0.8) = 32 / (-0.8); simplify *C*= -40, as expected.

We can also plot both temperature scales and see where they cross.

## Temperature in Kelvin

Another temperature scale is the Kelvin, which finds use in scientific work. The relationship between the Kelvin and degrees Celsius is K - 273.15 = *C*. K = 0 is the lowest temperature value obtainable. K = 0 is often called **absolute zero**. Also, we don't write K with a degree symbol but simply write K.

### Kelvin to Celsius

Here's another example: what if we have 30 Kelvins, what temperature in degrees Celsius does this correspond to?

From K - 273.15 = *C*, we get:

*C*= K - 273.15 = 30 - 273.15 = -243.15o C.

### Kelvin to Fahrenheit

What if we wanted to convert Kelvins to degrees Fahrenheit? A two step process: first, convert Kelvins to degrees Celsius and then convert Celsius to Fahrenheit. For example, the surface temperature of the sun is estimated to be 5,777 K. What is the corresponding o F temperature?

Step 1: Convert K to *C* using *C* = K - 273.15

*C* = K - 273.15 = 5,777 - 273.15 = 5,503.85

Step 2: Convert *C* to *F* using *F* = 1.8*C* + 32

*F* = 1.8*C* + 32 = 1.8(5,503.85) + 32 = 9,938.93

Planning a trip to Vancouver is fine, but there are no plans for a trip to the surface of the sun any time soon!

## Lesson Summary

All right, let's take a moment or two to review. As we learned, temperatures are typically measured in **degrees Celsius**, which is the temperature measurement in Canada and most of the rest of the world, and in **degrees Fahrenheit**, with is the temperature measurement in the United States.

At sea level, water freezes at 0o C which is 32o F. The boiling point of water is 100o C, which is 212o F. The equation relating the two scales is *F* = 1.8*C* + 32.

Scientific applications often use the Kelvin, where zero Kelvins is **absolute zero** temperature, which corresponds to *C* - 273.15.

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## Solving Problems With Temperature

Planning a trip to Vancouver from Boston includes gathering lots of information. It helps to know local temperatures while deciding on the appropriate clothing to bring. Temperatures in Canada are in **degrees Celsius** while in the United States, the temperature is measured in **degrees Fahrenheit**.

With Internet resources, these conversions are easily done but have you ever wondered how to do them yourself? With one simple equation, the conversions with Celsius and Fahrenheit temperatures are easily made. Have you ever wondered where this equation came from? Then there's the temperature measure called the Kelvin. In this lesson, we will see how they are all related.

## The Celsius-Fahrenheit Equation

The temperature in Boston this fine summer morning is 72o F while at the same time in Vancouver, the people are enjoying 15o C. To get to the same temperature scale, whether it's degrees Celsius or degrees Fahrenheit, means that these measurements have a relationship. We can convert from one to the other.

At sea level, water boils at 212o F and freezes at 32o F. That's a difference in temperatures of 212 - 32 = 180. You could say Î”F = 180. On the Celsius scale, water boils at 100o C and freezes at 0o C; a temperature difference Î”C = 100.

Let's think about a slope, *m* = Î”F / Î”C. The slope is Î”F / Î”C = 180/100 = 1.8.

Remember the equation of a line expressed as *y* = *mx* + *b* ? Well, what if we had *F* = *mC* + *b*? So, *F* = 1.8*C* + *b*.

What about the constant *b*? When water freezes at *C* = 0 we have *F* = 32. In our equation, *F* = 1.8*C* + *b*, let *C* = 0 and *F* = 32 to get 32 = 1.8(0) + *b*. Meaning, 32 = 0 + *b* or *b* = 32.

We're done! The beautiful conversion equation between *C* and *F* is:

*F*= 1.8*C*+ 32.

### Example 1

Let's look at an example. Our friends in Boston would like to know what a Vancouver temperature of 15o C is in degrees Fahrenheit.

From *F* = 1.8*C* + 32 let *C* = 15. Then, *F* = 1.8(15) + 32 = 59. Therefore, in Boston we have 72o F while in Vancouver it's 59o F. Big difference! 72 - 59 = 13 degrees.

### Example 2

Here's another example: our friends in Vancouver would like to know what the Boston temperature of 72o F is in Celsius.

From *F* = 1.8*C* + 32 , solve for *C* by subtracting 32 from both sides:

*F*- 32 = 1.8*C*+ 32 - 32; which simplifies to*F*- 32 = 1.8*C*. Then divide both sides by 1.8- (
*F*- 32) / 1.8 = 1.8*C*/ 1.8; which simplifies to - (
*F*- 32) / 1.8 =*C*

So, *C* = (*F* - 32) / 1.8 = (72 - 32) / 1.8 â‰… 22.

Boston is at 22o C while Vancouver is at 15o C. Only a 7o C change. The difference in temperature sounds better in degrees Celsius.

### Example 3

How about calculating the temperature in o F for another place in North America? How about Death Valley, CA,? Now here's a third example: the measured temperature in Death Valley is currently 35o C. What is this temperature in degrees Fahrenheit?

In degrees Fahrenheit this is *F* = 1.8*C* + 32 = 1.8(35) + 32 = 95.

Not bad, 95o F. However, on July 10, 1913, the hottest temperature ever recorded in the world was in Death Valley. This temperature was 134o F.

### Example 4

Here's one more example: our friends in Canada can't relate to this 134o F. Let's convert to degrees Celsius.

*C* = (*F* - 32) / 1.8 = (134 - 32) / 1.8 â‰… 57o C. More than half way up the Celsius scale from freezing towards boiling. No wonder it's called Death Valley!

## The Crossing Point

Is there a temperature value where *C* and *F* are the same? Turns out there is, and this value is -40. So if the thermometer reads -40o C in Vancouver, and it reads -40o F in Boston, the temperatures are the same.

To show this, we take the *F* = 1.8*C* + 32 equation and replace *F* with *C*. Then:

*C*= 1.8*C*+ 32. Subtract 1.8*C*from both sides:*C*- 1.8*C*= 1.8*C*+ 32 - 1.8*C*; then you simplify- -0.8
*C*= 32; divide both sides by -0.8: - -0.8
*C*/ (-0.8) = 32 / (-0.8); simplify *C*= -40, as expected.

We can also plot both temperature scales and see where they cross.

## Temperature in Kelvin

Another temperature scale is the Kelvin, which finds use in scientific work. The relationship between the Kelvin and degrees Celsius is K - 273.15 = *C*. K = 0 is the lowest temperature value obtainable. K = 0 is often called **absolute zero**. Also, we don't write K with a degree symbol but simply write K.

### Kelvin to Celsius

Here's another example: what if we have 30 Kelvins, what temperature in degrees Celsius does this correspond to?

From K - 273.15 = *C*, we get:

*C*= K - 273.15 = 30 - 273.15 = -243.15o C.

### Kelvin to Fahrenheit

What if we wanted to convert Kelvins to degrees Fahrenheit? A two step process: first, convert Kelvins to degrees Celsius and then convert Celsius to Fahrenheit. For example, the surface temperature of the sun is estimated to be 5,777 K. What is the corresponding o F temperature?

Step 1: Convert K to *C* using *C* = K - 273.15

*C* = K - 273.15 = 5,777 - 273.15 = 5,503.85

Step 2: Convert *C* to *F* using *F* = 1.8*C* + 32

*F* = 1.8*C* + 32 = 1.8(5,503.85) + 32 = 9,938.93

Planning a trip to Vancouver is fine, but there are no plans for a trip to the surface of the sun any time soon!

## Lesson Summary

All right, let's take a moment or two to review. As we learned, temperatures are typically measured in **degrees Celsius**, which is the temperature measurement in Canada and most of the rest of the world, and in **degrees Fahrenheit**, with is the temperature measurement in the United States.

At sea level, water freezes at 0o C which is 32o F. The boiling point of water is 100o C, which is 212o F. The equation relating the two scales is *F* = 1.8*C* + 32.

Scientific applications often use the Kelvin, where zero Kelvins is **absolute zero** temperature, which corresponds to *C* - 273.15.

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- Activities
- FAQs

## Temperature Matching Game:

### Reminders:

- To convert Fahrenheit to Celsius, we use the formula
*F*= 1.8*C*+ 32. - To convert Celsius to Fahrenheit, we use the formula
*C*= (*F*- 32) / 1.8. - To convert Kelvin to Celsius, we use the formula
*C*=*K*- 273.15. - To covert Celsius to Kelvin, we use the formula
*K*=*C*+ 273.15.

### Materials Needed:

- Notecards with temperatures on them, such that each notecard with a Fahrenheit temperature on it has a corresponding notecard in the bunch with an equivalent temperature in Celsius and a corresponding notecard in the bunch with an equivalent temperature in Kelvin. The number of notecards needed will be the largest number that is divisible by 3, but less than or equal to the number of students in the class. This is because corresponding notecards will be in groups of 3.

### How to Play One Round:

- Have each student in the class draw a card from the group of notecards. If the number of students in the class is not divisible by 3, they can take turns being the judges while they sit around out of the game. Otherwise, the teacher can be the judge.
- Each student must find the two students that have cards with temperatures that are equivalent to theirs.
- The three students that find their matching group first win the round. The judge will decide on who was first if there are any close outcomes.

### Discussion Questions:

- Suppose Mary got the notecard with 15° Celsius on it. What will be on the cards of the others in her matching group?
- Suppose Joy got the notecard with 318.15° Kelvin on it, and Mike got the notecard with 40° Celsius on it. Will Joy and Mike be in the same matching group? Why or why not?

### Answers:

- 15° Celsius is equivalent to 59° Fahrenheit and 288.15° Kelvin, so these will be the values on the cards of the others in Mary's matching group.
- No, because 318.15° Kelvin is equivalent to 45° Celsius, not 40° Celsius, so their cards are not equivalent. Hence, they won't be in the same matching group.

#### What are the temperature conversion formulas?

To convert from Fahrenheit to Celsius:

T (°C) = ( T (°F) - 32 )/1.8

To convert from Celsius to Fahrenheit:

T (°F) = ( 1.8 * T (°C) ) + 32

#### What is the Celsius value for 98.6 F?

Using the following formula to convert the temperature from Fahrenheit to Celsius:

T (°C) = ( T (°F) - 32 )/1.8

= (98.6°F - 32)/1.8 = 37°C

#### What does 40 Celsius mean in Fahrenheit?

Convert from Celsius to Fahrenheit:

T (°F) = ( 1.8 * T (°C) ) + 32

T (°F) = ( 1.8 * 40 (°C) ) + 32 = 104 °F

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