# Metric Units of Length Conversion

 $\require{cancel}$ $\require{bbox}$

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

1 decameter = 1000 × 10 × 10 × 10 × 10 micrometers = 10000000 micrometers

## 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 $\text{cm}$

Example 4
Convert $12500 \; \text{mm}$ to $\text{hm}$

Example 5
Convert $0.0023 \; \text{cm}$ into $\text{nm}$

Example 6
Convert $890000 \; \mu\text{m}$ to $\text{hm}$

## Exercises

Part A
Use table 1 above to determine the factors of conversion and convert

1. $\quad 0.0102 \text { Mm}$ to $\text{hm}$,
2. $\quad 0.3 \text { nm}$ to $\text{mm}$,
3. $\quad 1200 \text { m}$ to $\text{Mm}$,
4. $\quad 13200 \text { dm}$ to $\text{km}$,
5. $\quad 0.002 \text { Gm}$ to $\text{km}$,
6. $\quad 13200 { \; \mu} \text m$ to $\text{dm}$,
7. $\quad 0.003 \text { dam}$ to $\text{mm}$,
8. $\quad 0.001 \text { cm}$ to $\text{nm}$,