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Derived Units

A combination of SI base units. Example: Joules.

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Derived Units

Picture of a farm

Credit: User:Joegrimes/Wikipedia
Source: http://commons.wikimedia.org/wiki/File:Farmlandlysander.JPG
License: CC BY-NC 3.0

How has farming evolved?

As farming becomes more expensive and less profitable (at least for small farms), many families will sell the land to builders who want to erect either commercial or residential properties. In order to sell, an accurate property tile is needed. The dimensions of the farm must be determined and the acreage calculated from those dimensions.

Dimensional Analysis and Derived Units

Some units are combinations of SI base units. A derived unit is a unit that results from a mathematical combination of SI base units. We have already discussed volume and energy as two examples of derived units.  Some others are listed in Table below:

Derived SI Units
Quantity Symbol Unit Unit Abbreviation Derivation
Area A square meter m2 length × width
Volume V cubic meter m3 length × width × height
Density D kilograms/cubic meter kg/m3 \begin{align*}\frac{\text{mass}}{\text{volume}}\end{align*}
Concentration c moles/liter mol/L \begin{align*}\frac{\text{amount}}{\text{volume}}\end{align*}
Speed (velocity) v meters/second m/s \begin{align*}\frac{\text{length}}{\text{time}}\end{align*}
Acceleration a meters/second/second m/s2 \begin{align*}\frac{\text{speed}}{\text{time}}\end{align*}
Force F newton N mass × acceleration
Energy E joule J force × length

Using dimensional analysis with derived units requires special care. When units are squared or cubed as with area or volume, the conversion factors themselves must also be squared. Shown below is the conversion factor for cubic centimeters and cubic meters.

\begin{align*}\left(\frac{1 \ \text{m}}{100 \ \text{cm}}\right)^3=\frac{1 \ \text{m}^3}{10^6 \ \text{cm}^3}=1\end{align*}

Because a cube has 3 sides, each side is subject to the conversion of 1 m to 100 cm. Since 100 cubed is equal to 1 million (106), there are 106 cm3 in 1 m3. Two convenient volume units are the liter, which is equal to a cubic decimeter, and the milliliter, which is equal to a cubic centimeter. The conversion factor would be:

\begin{align*}\left(\frac{1 \ \text{dm}}{10 \ \text{cm}}\right)^3 = \frac{1 \ \text{dm}^3}{1000 \ \text{cm}^3}=1\end{align*}

There are thus 1000 cm3 in 1 dm3, which is the same thing as saying there are 1000 mL in 1 L

1000 milliliter cubes are in a liter cube

Credit: CK-12 Foundation - Christopher Auyeung
License: CC BY-NC 3.0

There are 1000 cm3 in 1 dm3. Since a cm3 is equal to a mL and a dm3 is equal to a L, we can say that there are 1000 mL in 1 L.[Figure2]

Sample Problem: Derived Unit Conversion

Convert 3.6 × 108 mm3 to mL.

Step 1: List the known quantities and plan the problem.


  • 1 m = 1000 mm
  • 1 ml = 1 cm3
  • 1 m = 100 cm


  • 3.6 mm3 = ? mL

This problem requires multiple steps and the technique for converting with derived units.  Simply proceed one step at a time: mm3 to m3 to cm3 = mL.

Step 2: Calculate

\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{m}}{1000 \ \text{mm}}\right)^3 \times \left(\frac{100 \ \text{cm}}{1 \ \text{m}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}

Numerically, the steps are to divide 3.6 by 109, followed by multiplying by 106.  You may find that you can shorten the problem by a step by first determining the conversion factor from mm to cm and using that instead of first converting to m. There are 10 mm in 1 cm.

\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{cm}}{10 \ \text{mm}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}

In this case 3.6 / 1000 gives the same result of 0.0036.

Step 3: Think about your result.

Cubic millimeters are much smaller than cubic centimeters, so the final answer is much less than the original number of mm3.


  • A derived unit is a unit that results from a mathematical combination of SI base units.
  • Calculations involving derived units follow the same principles as other unit conversion calculations.


  1. What is a derived unit?
  2. Convert 0.00722 km2 to m2.
  3. Convert 129 cm3 to L.
  4. Convert 4.9 × 105 μm3 to mm3.

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Image Attributions

  1. [1]^ Credit: User:Joegrimes/Wikipedia; Source: http://commons.wikimedia.org/wiki/File:Farmlandlysander.JPG; License: CC BY-NC 3.0
  2. [2]^ Credit: CK-12 Foundation - Christopher Auyeung; License: CC BY-NC 3.0

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