Table and Factor of Conversion of Units of Area
The table shown below helps in finding factors of conversion between metric units of length which will be used to find the factor of conversion of areas.
For example, using the table above we can use the factors below the arrows to write:
\[ 1 \text{ dam} = 10 \times 10 \text{ dm} = 100 \text{ dm} \]which gives:
\[ 1 \text{ dam} = 100 \text{ dm} \]Multiply each side of the above equality by itself (or square both sides):
\[ (1 \text{ dam})(1 \text{ dam}) = (100 \text{ dm})(100 \text{ dm}) \]Simplify to write:
\[ 1 \text{ dam}^2 = 10,000 \text{ dm}^2 \]which gives the following factors of conversion between the units of area \( \text{dam}^2 \) and \( \text{dm}^2 \):
\[ \displaystyle \frac{1 \text{ dam}^2}{10,000 \text{ dm}^2} = 1 \]or
\[ \displaystyle \frac{10,000 \text{ dm}^2}{1 \text{ dam}^2} = 1 \]Note that the factor of conversion may be written in two different ways by interchanging the denominator and numerator. We select the factor of conversion with the unit in the denominator that is the same as the given unit to be converted so that they cancel.
Examples of Conversion with Solutions
In the first few examples, we show all the steps for a thorough understanding of the conversion. Apart from the table of metric units above, nothing else, such as formulas for example, is needed to do the conversions.
Example 1
Convert \( 2450 \text{ cm}^2 \) to \( \text{m}^2 \)
View Solution
We are converting \( \text{cm}^2 \) to \( \text{m}^2 \) and therefore the factor of conversion between \( \text{cm} \) and \( \text{m} \) is needed first.
Using Table 1 above, we have:
\[ 1 \text{ m} = 100 \text{ cm} \]Square both sides:
\[ (1 \text{ m})^2 = (100 \text{ cm})^2 \]Simplify:
\[ 1 \text{ m}^2 = 10,000 \text{ cm}^2 \]which gives the factors of conversion:
\[ \displaystyle \frac{1 \text{ m}^2}{10,000 \text{ cm}^2} = 1 \quad (I) \]or
\[ \displaystyle \frac{10,000 \text{ cm}^2}{1 \text{ m}^2} = 1 \quad (II) \]To convert \( 2450 \text{ cm}^2 \), we use the factor of conversion (I) given by \( \displaystyle \frac{1 \text{ m}^2}{10,000 \text{ cm}^2} \) because it has \( \text{cm}^2 \) in the denominator which will cancel with the given \( \text{cm}^2 \).
Write the given expression as:
\[ 2450 \text{ cm}^2 = 2450 \text{ cm}^2 \times \color{red}1 \]Substitute \( \color{red}1 \) by the factor of conversion \( \displaystyle \frac{1 \text{ m}^2}{10,000 \text{ cm}^2} \):
\[ 2450 \text{ cm}^2 = 2450 \text{ cm}^2 \times \displaystyle \frac{1 \text{ m}^2}{10,000 \text{ cm}^2} \]Cancel \( \text{cm}^2 \):
\[ 2450 \text{ cm}^2 = 2450 \cancel{\text{ cm}^2} \times \displaystyle \frac{1 \text{ m}^2}{10,000 \cancel{\text{ cm}^2}} \]Simplify:
\[ 2450 \text{ cm}^2 = \frac{2450 \times 1}{10,000} \text{ m}^2 \]Evaluate:
Example 2
Convert \( 34,590.5 \text{ m}^2 \) to \( \text{km}^2 \)
View Solution
We are converting \( \text{m}^2 \) to \( \text{km}^2 \) and therefore the factor of conversion between \( \text{m} \) and \( \text{km} \) is needed first.
Use the table above to write the factor of conversion between \( \text{m} \) and \( \text{km} \):
\[ 1 \text{ km} = 1000 \text{ m} \]Square both sides:
\[ (1 \text{ km})(1 \text{ km}) = (1000 \text{ m})(1000 \text{ m}) \]and simplify:
\[ 1 \text{ km}^2 = 1,000,000 \text{ m}^2 \]which gives the factor of conversion of area with \( \text{m}^2 \) in the denominator:
\[ \displaystyle \frac{1 \text{ km}^2}{1,000,000 \text{ m}^2} = 1 \]Convert using the above factor of conversion:
\[ 34,590.5 \text{ m}^2 = 34,590.5 \text{ m}^2 \times \displaystyle \frac{1 \text{ km}^2}{1,000,000 \text{ m}^2} \]Cancel \( \text{m}^2 \):
\[ 34,590.5 \text{ m}^2 = 34,590.5 \cancel{\text{ m}^2} \times \displaystyle \frac{1 \text{ km}^2}{1,000,000 \cancel{\text{ m}^2}} \]Simplify:
\[ 34,590.5 \text{ m}^2 = \frac{34,590.5 \times 1}{1,000,000} \text{ km}^2 \]Evaluate:
Questions with Solutions
Convert the following:
- \( 569,000 \text{ mm}^2 \) to \( \text{m}^2 \)
- \( 1.2 \text{ km}^2 \) to \( \text{dam}^2 \)
- \( 23.01 \text{ hm}^2 \) to \( \text{m}^2 \)
- \( 12.7 \text{ cm}^2 \) to \( \text{mm}^2 \)
- \( 13,500 \text{ dm}^2 \) to \( \text{hm}^2 \)
Solutions to the Above Questions
-
Convert \( 569,000 \text{ mm}^2 \) to \( \text{m}^2 \)
Use Table 1 to write:
\[ 1 \text{ m} = 1000 \text{ mm} \]Square both sides:
\[ 1 \text{ m}^2 = 1,000,000 \text{ mm}^2 \]Factor of conversion:
\[ \displaystyle \frac{1 \text{ m}^2}{1,000,000 \text{ mm}^2} = 1 \]Multiply and simplify:
\[ 569,000 \text{ mm}^2 \times \frac{1 \text{ m}^2}{1,000,000 \text{ mm}^2} = \frac{569,000}{1,000,000} \text{ m}^2 \]Evaluate:
\[ 569,000 \text{ mm}^2 = 0.569 \text{ m}^2 \] -
Convert \( 1.2 \text{ km}^2 \) to \( \text{dam}^2 \)
Use Table 1 to write:
\[ 1 \text{ km} = 100 \text{ dam} \]Square both sides:
\[ 1 \text{ km}^2 = 10,000 \text{ dam}^2 \]Factor of conversion:
\[ \displaystyle \frac{10,000 \text{ dam}^2}{1 \text{ km}^2} = 1 \]Multiply and simplify:
\[ 1.2 \text{ km}^2 \times \frac{10,000 \text{ dam}^2}{1 \text{ km}^2} = 1.2 \times 10,000 \text{ dam}^2 \]Evaluate:
\[ 1.2 \text{ km}^2 = 12,000 \text{ dam}^2 \] -
Convert \( 23.01 \text{ hm}^2 \) to \( \text{m}^2 \)
Use Table 1 to write:
\[ 1 \text{ hm} = 100 \text{ m} \]Square both sides:
\[ 1 \text{ hm}^2 = 10,000 \text{ m}^2 \]Factor of conversion:
\[ \displaystyle \frac{10,000 \text{ m}^2}{1 \text{ hm}^2} = 1 \]Multiply and simplify:
\[ 23.01 \text{ hm}^2 \times \frac{10,000 \text{ m}^2}{1 \text{ hm}^2} = 23.01 \times 10,000 \text{ m}^2 \]Evaluate:
\[ 23.01 \text{ hm}^2 = 230,100 \text{ m}^2 \] -
Convert \( 12.7 \text{ cm}^2 \) to \( \text{mm}^2 \)
Use Table 1 to write:
\[ 1 \text{ cm} = 10 \text{ mm} \]Square both sides:
\[ 1 \text{ cm}^2 = 100 \text{ mm}^2 \]Factor of conversion:
\[ \displaystyle \frac{100 \text{ mm}^2}{1 \text{ cm}^2} = 1 \]Multiply and simplify:
\[ 12.7 \text{ cm}^2 \times \frac{100 \text{ mm}^2}{1 \text{ cm}^2} = 12.7 \times 100 \text{ mm}^2 \]Evaluate:
\[ 12.7 \text{ cm}^2 = 1270 \text{ mm}^2 \] -
Convert \( 13,500 \text{ dm}^2 \) to \( \text{hm}^2 \)
Use Table 1 to write:
\[ 1 \text{ hm} = 1000 \text{ dm} \]Square both sides:
\[ 1 \text{ hm}^2 = 1,000,000 \text{ dm}^2 \]Factor of conversion:
\[ \displaystyle \frac{1 \text{ hm}^2}{1,000,000 \text{ dm}^2} = 1 \]Multiply and simplify:
\[ 13,500 \text{ dm}^2 \times \frac{1 \text{ hm}^2}{1,000,000 \text{ dm}^2} = \frac{13,500}{1,000,000} \text{ hm}^2 \]Evaluate:
\[ 13,500 \text{ dm}^2 = 0.0135 \text{ hm}^2 \]