# Convert Metric Units of Area

\( \)\( \)\( \)\( \) \( \require{cancel} \) \( \require{bbox} \)Convert metric units of area, such as, \( \text{m}^2, \text{cm}^2, \text{mm}^2, \text{km}^2, ..\), examples with solutions are presented including more More questions with solutions .

## 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) of the above equality \[ (1 \text{ dam})(1 \text{ dam}) = (100 \text{ dm})(100 \text{ dm}) \] Simplify to write \[ 1 \text{ dam}^2 = 10000 \text{ dm}^2 \] which gives the following factors of conversion between the units of area \( \text{ dam}^2 \) and \( \text{ dm}^2 \). \[ \displaystyle \frac{1 \text{ dam}^2}{10000 \text{ dm}^2} = 1\] or \[ \displaystyle \frac {10000 \text{ dm}^2} {1 \text{ dam}^2} = 1\] 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 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 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 \( \text{m}^2 \)

__ Solution to Example 1 __

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 \[ 1 \text{ m}^2 = 10000 \text{ cm}^2 \] which gives the factors of conversion \[ \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} = 1 \quad (I) \] or \[ \displaystyle \frac{10000 \text{ cm}^2}{1 \text{ m}^2} = 1 \quad (II) \] To convert \( 2450 \text{ cm}^2 \), we use use the factor of conversion (I) given by \( \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \) 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 \( \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \) since it is equal to \( \color{red}1 \)

\( 2450 \text{ cm}^2 = 2450 \text{ cm}^2 \times \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \)

Cancel \( \text{ cm}^2 \)

\( 2450 \text{ cm}^2 = 2450 \cancel{\text{ cm}^2} \times \displaystyle \frac{1 \text{ m}^2}{10000 \cancel{\text{ cm}^2}} \)

Simplify

\( 2450 \text{ cm}^2 = \displaystyle \frac{2450 \times 1 }{10000} \text{ m}^2 \)

Evaluate

\[ \bbox[10px, border: 2px solid red] { 2450 \text{ cm}^2 = 0.245 \text{ m}^2 } \]

Using Table 1 above, we have \[ 1 \text{ m} = 100 \text{ cm}\] Square both sides \[ (1 \text{ m})^2 = (100 \text{ cm})^2 \] Simplify \[ 1 \text{ m}^2 = 10000 \text{ cm}^2 \] which gives the factors of conversion \[ \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} = 1 \quad (I) \] or \[ \displaystyle \frac{10000 \text{ cm}^2}{1 \text{ m}^2} = 1 \quad (II) \] To convert \( 2450 \text{ cm}^2 \), we use use the factor of conversion (I) given by \( \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \) 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 \( \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \) since it is equal to \( \color{red}1 \)

\( 2450 \text{ cm}^2 = 2450 \text{ cm}^2 \times \displaystyle \frac{1 \text{ m}^2}{10000 \text{ cm}^2} \)

Cancel \( \text{ cm}^2 \)

\( 2450 \text{ cm}^2 = 2450 \cancel{\text{ cm}^2} \times \displaystyle \frac{1 \text{ m}^2}{10000 \cancel{\text{ cm}^2}} \)

Simplify

\( 2450 \text{ cm}^2 = \displaystyle \frac{2450 \times 1 }{10000} \text{ m}^2 \)

Evaluate

\[ \bbox[10px, border: 2px solid red] { 2450 \text{ cm}^2 = 0.245 \text{ m}^2 } \]

Example 2

Convert \( 34590.5 \text{ m}^2 \) to \( \text{km}^2 \)

__ Solution to Example 2 __

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 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 \[ 1 \text{ km}^2 = 1000000 \text{ m}^2 \] which gives the factors of conversion of area with \( \text{ m}^2 \) in the denominator \[ \displaystyle \frac{1 \text{ km}^2}{1000000 \text{ m}^2} = 1 \] Convert using the above factor of conversion

\( 34590.5 \text{ m}^2 = 34590.5 \text{ m}^2 \times \displaystyle \frac{1 \text{ km}^2}{1000000 \text{ m}^2} \)

Cancel \( \text{ m}^2 \)

\( 34590.5 \text{ m}^2 = 34590.5 \cancel{\text{ m}^2} \times \displaystyle \frac{1 \text{ km}^2}{1000000 \cancel{\text{ m}^2}} \)

Simplify

\( 34590.5 \text{ m}^2 = \displaystyle \frac{34590.5 \times 1 }{1000000} \text{ km}^2 \)

Evaluate

\[ \bbox[10px, border: 2px solid red] { 34590.5 \text{ m}^2 = 0.0345905 \text{ km}^2 } \]

Use the table above to write 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 \[ 1 \text{ km}^2 = 1000000 \text{ m}^2 \] which gives the factors of conversion of area with \( \text{ m}^2 \) in the denominator \[ \displaystyle \frac{1 \text{ km}^2}{1000000 \text{ m}^2} = 1 \] Convert using the above factor of conversion

\( 34590.5 \text{ m}^2 = 34590.5 \text{ m}^2 \times \displaystyle \frac{1 \text{ km}^2}{1000000 \text{ m}^2} \)

Cancel \( \text{ m}^2 \)

\( 34590.5 \text{ m}^2 = 34590.5 \cancel{\text{ m}^2} \times \displaystyle \frac{1 \text{ km}^2}{1000000 \cancel{\text{ m}^2}} \)

Simplify

\( 34590.5 \text{ m}^2 = \displaystyle \frac{34590.5 \times 1 }{1000000} \text{ km}^2 \)

Evaluate

\[ \bbox[10px, border: 2px solid red] { 34590.5 \text{ m}^2 = 0.0345905 \text{ km}^2 } \]

## Questions with Solutions

Convert the following

- \( 569000 \text{ mm}^2 \) 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 \( \text{mm}^2 \)
- \( 13500 \text{ dm}^2 \) to \( \text{hm}^2 \)

## More References and links