Unit conversion can be tricky! There are several famous examples of unit conversions leading to huge mistakes, some of which were very costly! In 1999, NASA lost $125 million dollars as the Mars Climate Orbiter drifted off course and was lost due to unit conversion errors. NASA and Lockheed Martin failed to communicate the units of their shared data and calculations, which resulted in the lost rover.

So it's a good idea to learn how to be careful with units and ensure no mistakes like this happen to you.

## What is Unit Conversion?

Most numerical quantities in life have dimensions. Sometimes that dimension is a length, a mass, a volume or any other general type of quantity. The specific way that we can describe a dimension is a unit. Therefore a quantity that has a dimension of length can have units of meters, miles, feet, kilometers, inches and so forth. Unit conversion is the process of switching between units.

Remember, you can switch between units, but you cannot switch between dimensions. This means that a quantity with dimensions of length or area cannot be converted to a volume: the process of determining volume from length or area is not a conversion but an arithmetic operation. While conversions also involve arithmetic, this is a subtle difference worth remembering.

## Centimeters to Cubic Meters (cm to m^{3})?

Based on what we just discussed, we now know that you cannot use unit conversion to convert centimeters to cubic meters: centimeters is a unit of length and cubic meters is a unit of volume. Similarly, you can't convert centimeters to cubic centimeters or a meter to a cubic meter. However, we can convert centimeters to meters, and determine how many cubic centimeters are in a cubic meter.

In the metric system, the base unit is a meter, with any prefix added to meter telling us a scale. A centimeter is 1/100th of a meter, meaning that there are 100 centimeters in one meter. From this, we can quickly convert between centimeters and meters: 1 meter is 100 centimeters and 50 centimeters is 0.5 meters.

## General Unit Conversion

Unit conversion relies on multiplying the quantity that you are trying to convert by the number one. This sounds odd but let's talk it out!

Multiplying a number by one doesn't change it's value – correct? We can also express the number one in a lot of ways: 5/5 = 1 and 100000/100000 = 1. See a pattern? If the numerator and denominator are equal to each other, then one divided by the other is a one. So since 100 centimeters equals 1 meter, 100 cm/1 m = 1.

This is how you can build conversion factors that allow you to convert a quantity from one unit to another. If you have a quantity in centimeters, to convert it to meters you have to divide by the conversion factor such that the centimeter units cancel out and leave meters. If you need to know how many centimeters are in 5 meters, multiply by the conversion factor: 5 m **×** 100 cm / 1 m = 500 cm.

## Length to Volume

A volume describes the amount of space a three dimensional object occupies. Therefore, it has three dimensions. You can describe the sides of a box in three different units (length, *L*, width, *W* and height, *H*) of length, such as 14 inches, 1.5 feet, and 56 centimeters. The volume is

but if we don't convert units then our volume has units of inches feet centimeters, which is not very useful.

The simplest way to approach this is to convert two of the measurements to the same units as the third. Since we want to calculate cubic meters, let's determine each measurement in centimeters. We know there are 12 inches in one foot, so our length and width are 14 inches and (1.5 ft × 12 inch/ft = 18 inches). There are 2.54 cm in one inch, so converting again gives: 14 inch × 2.54 cm/in = 35.56 cm and 18 inch × 2.54 cm/in = 45.72 cm.

Using our conversion factor from earlier and the volume formula, we know that the volume of the box is 0.09 cubic meters.

References

About the Author

Lipi Gupta is currently pursuing her Ph. D. in physics at the University of Chicago. She earned her Bachelor of Arts in physics with a minor in mathematics at Cornell University in 2015, where she was a tutor for engineering students, and was a resident advisor in a first-year dorm for three years. With this experience, when not working on her Ph. D. research, Gupta participates in STEM outreach activities to promote young women and minorities to pursue science careers.

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