Examples of conversion between metric units of length are presented. A table based on the SI prefixes used with units is used to convert between metric units of length such as centimeters, millimeters, meters, ....
More exercises with solutions are presented at the bottom of the page.
Table of Conversion
The table of conversion is shown below. The factors under the arrows are fators of multiplications in going from a given unit to another unit on its left.
For example in going from millimeter to centimeter , there is a factor of 10 under the arrow, hence
1 centimeter = 10 millimeters
Another example, in going from micrometer to decameter , the factors under the arrows are :
1000 (from micrometer to millimeter) ,
10 (from millimeter to centimeter),
10 (from centimeter to decimeter),
10 (from decimeter to meter),
10 (from meter to decameter),
We now combine all factors and write
Table. 1 - Metric Units of Length Conversion Table
Factors of Conversion From the Table of Conversion
Table 1 above is used to determine the factor of conversion in order to conver from one unit to another.
Example 1
Convert 2.1 km (kilometer) into dam (decameter).
Solution to Example 1
In what follows, we use abbreviation \( \text { km} \) for kilometers and the abbreviation \( \text{ dam} \) for decameter.
Using the factors in Table 1 above, kilometer which is to the left of decameter, is given by
\[ 1 \text { km} = 10 \times 10 \text{ dam} = 100 \text{ dam} \]
which may be written as
\[ 1 \text { km} = 100 \text{ dam} \]
The above equality gives the following factor of conversion
\[ \displaystyle \frac{1 \text { km}}{100 \text{ dam}} = 1 \quad (I) \]
and
\[ \displaystyle \frac{100 \text{ dam}} {1 \text { km}} = 1 \quad (II) \]
The above factors of conversion are used to convert \( 2.1 \text { km} \) to \( \text { dam} \).
Write
\[ 2.1 \text{ km} = 2.1 \text{ km} \times 1 \]
Substitute \( 1 \) by \( \displaystyle \frac{100 \text{ dam}} {1 \text { km}} \) which is the the factor of conversion (II) found above because it is equal to 1
\[ 2.1 \text{ km} = 2.1 \text{ km} \times \displaystyle \frac{100 \text{ dam}} {1 \text { km}} \]
Note that we have used factor (II) with \( \text { km} \) in the denominator so that it cancels with the \( \text { km} \) in the given \( 2.1 \text{ km} \)
Cancel \( \text { km} \) on the right
\[ 2.1 \text{ km} = 2.1 \cancel{\text{ km}} \times \displaystyle \frac{100 \text{ dam}} {1 \cancel{\text{ km}}} \]
Simplify and evaluate
\[ 2.1 \text{ km} = 2.1 \times \displaystyle \frac{100 \text{ dam}} {1} \]
\[ \bbox[10px, border: 2px solid red] {2.1 \text{ km} = \displaystyle \frac{ 2.1 \times 100 \text{ dam}} {1} = 210 \text{ dam} } \]
Example 2
Convert \( 1200000 \text{ nm} \) (nanometer) into \( \text{ dm} \) (decimeter).
Solution to Example 2
Nanometer \( \text{ nm} \) is smaller than decimeter \( \text{ dm} \) and therefore using the factors in Table 1 above going from \( \text{ nm} \) to \( \text{ dm} \) , we write
\[ 1 \text { dm} = 1000 \times 1000 \times 10 \times 10 = 100000000 \text{ nm} \]
which may be written as
\[ 1 \text { dm} = 100000000 \text{ nm} \]
From the above we can write two factors of conversion
\[ \displaystyle \frac{1 \text { dm}}{100000000 \text{ nm}} = 1 \quad (I) \]
and
\[ \displaystyle \frac{100000000 \text{ nm}} {1 \text { dm}} = 1 \quad (II) \]
Since we are given \( \text{ nm} \) and we need to cancel them, we use the factor of conversion (I) because it has \( \text{ nm} \) in the denominator which may be canceled.
The question in this example is to convert 1200000 \( \text{ nm} \) to \( \text{ dm} \). Hence
\[ 1200000 \text{ nm} = 1200000 \text{ nm} \times \displaystyle \frac{1 \text { dm}}{100000000 \text{ nm}} \]
Cancel \( \text{ nm} \) on the right
\[ 1200000 \text{ nm} = 1200000 \cancel{\text{ nm}} \times \displaystyle \frac{1 \text { dm}}{100000000 \cancel{\text{ nm}}} \]
Simplify and evaluate
\[ 1200000 \text{ nm} = \displaystyle \frac{ 1200000 \times 1 \text { dm}}{100000000 } \]
\[ \bbox[10px, border: 2px solid red]{ 1200000 \text{ nm} = 0.012 \text{ dm} } \]
Examples with Solutions
Example 3
Convert \( 170 \; \text{dam} \) into \( \text{cm} \)
Solution to Example 3
Using the factors Table 1 above going from \( \text{cm} \) to \( \text{dam} \), we write
\[ 1 \text { dam} = 10 \times 10 \times 10 \text{ cm} = 1000 \text{ cm} \]
which gives the factors of conversion
\[ \displaystyle \frac{1 \text { dam}}{1000 \text{ cm}} = 1 \quad (I) \]
and
\[ \displaystyle \frac{1000 \text{ cm}} {1 \text { dam}} = 1 \quad (II) \]
We are given \( \text { dam} \) we therefore use the factor of conversion (II) since it has \( \text { dam} \) in the denominator which may be cancelled.
Convert \( 170 \; \text{dam} \) as follows
\[ 170 \; \text{dam} = 170 \; \text{dam} \times \displaystyle \frac{1000 \text{ cm}} {1 \text { dam}} \]
Cancel \( \text{dam} \)
\[ 170 \; \text{dam} = 170 \cancel{\text{dam}} \times 1000 \displaystyle \frac{\text{cm}}{\cancel{\text{dam}}} \]
Evaluate
\[ 170 \; \text{dam} = 170 \times 1000 \text{ cm} = 170000 \text{ cm}\]
Example 4
Convert \( 12500 \; \text{mm} \) to \( \text{hm} \)
Solution to Example 4
Using the factors from \( \text{mm} \) to \( \text{hm} \) in Table 1 above, we write
\[ 1 \text { hm} = 10 \times 10 \times 10 \times 10 \times 10 \text{ mm} = 100000 \text{ mm} \]
which gives the factors of conversion
\[ \displaystyle \frac{1 \text { hm}}{100000 \text{ mm}} = 1 \quad (I) \]
and
\[ \displaystyle \frac {100000 \text{ mm}} {1 \text{ hm}} = 1 \quad (II) \]
We are given \( \text{ mm} \) which needs to be canceled, we therefore use the factor of conversion (I), because it has \( \text{ mm} \) in the denominator.
Convert \( 12500 \; \text{mm} \) as follows
\[ 12500 \; \text{mm} = 12500 \text{ mm} \times \displaystyle \frac{1 \text { hm}}{100000 \text{ mm}} \]
Cancel \( \text{mm} \) on the right
\[ 12500 \; \text{mm} = 12500 \cancel{\text{mm}} \times \displaystyle \frac{1 \text { hm}}{100000 \cancel{\text{mm}}} \]
Simplify and evaluate
\[ 12500 \; \text{mm} = \displaystyle \frac{12500}{100000} \text{hm} \]
Divide
\[ 12500 \; \text{mm} = 0.125 \text{ hm}\]
Example 5
Convert \( 0.0023 \; \text{cm} \) into \( \text{nm} \)
Solution to Example 5
Using the factors Table 1 above from \( \text{nm} \) to \( \text{cm} \), we write
\[ 1 \text { cm} = 1000 \times 1000 \times 10 \text{ nm} = 10000000 \text{ nm} \]
which gives the factors of conversion
\[ \displaystyle \frac{1 \text { cm}}{10000000 \text{ nm}} = 1 \quad (I) \]
and
\[ \displaystyle \frac {10000000 \text{ nm}} {1 \text { cm}} = 1 \quad (II) \]
We are given \( \text{cm} \) which needs to be canceled and we therefore use the factor of conversion (II), because it has \( \text { cm} \) in the denominator.
Convert \( 0.0023 \; \text{cm} \) as follows
\[ 0.0023 \; \text{cm} = 0.0023 \; \text{cm} \times \times 10000000 \displaystyle \frac{\text{nm}}{\text{cm}} \]
Cancel \( \text{cm} \) on the right
\[ 0.0023 \; \text{ cm} = 0.0023 \cancel{\text{cm}} \times 10000000 \displaystyle \frac{\text{nm}}{\cancel{\text{cm}}}\]
Simplify and evaluate
\[ 0.0023 \; \text{ cm} = 0.0023 \times 10000000 \text{ nm} = 23000 \text{ nm}\]
Example 6
Convert \( 890000 \; \mu\text{m} \) to \( \text{hm} \)
Solution to Example 6
Using the factors Table 1 above from \( \; \mu\text{m} \) to and \( \text{ hm} \), we write
\[ 1 \text { hm} = 1000 \times 10 \times 10 \times 10 \times 10 \times 10 \; \mu\text{m} = 100000000 \; \mu\text{m} \]
which gives the factors of conversion
\[ \displaystyle \frac{1 \text { hm}}{100000000 \mu\text{m}} = 1 \quad (I) \]
and
\[ \displaystyle \frac{100000000 \mu\text{m}} {1 \text { hm}} = 1 \quad (II) \]
We are given \( \mu\text{m} \) which needs to be canceled and we therefore use the factor of conversion (I), because it has \( \mu\text{m} \) in the denominator.
Convert \( 890000 \; \mu\text{m} \) as follows
\[ 890000 \; \mu m = 890000 \; \mu m \times \displaystyle \frac{1}{100000000} \displaystyle \frac{\text{hm}}{\mu m} \]
Cancel \( \mu\text{m} \)
\[ 890000 \; \mu m = 890000 \; \cancel{\mu\text{m}} \times \displaystyle \frac{1}{100000000} \displaystyle \frac{\text{hm}}{\cancel{\mu\text{m}}} \]
Simplify and evaluate
\[ 890000 \; \mu m = \displaystyle \frac{890000 \times 1}{100000000} \text{ hm} = 0.0089 \text{ hm}\]
Exercises
Part A
Use table 1 above to determine the factors of conversion and convert