Unit Conversion

We often need to convert between units of area and volume. When we do this, we need to look out for some common mistakes.

Converting areas

A common source of error is forgetting to apply a linear conversion multiple times to get an area or volume conversion.

1 m^2 = 1 \cdot meter \cdot meter

Here we convert a square meter to square centimeters.

1 \cdot meter \cdot meter \cdot \frac{100cm}{meter} \cdot \frac{100cm}{meter} = 10^4 cm^2

We can also convert from mixed unit areas to other units.

Example: Convert 2 inch-feet to square centimeters

2\; \cancel{\textrm{inch}}\; \cancel{\textrm{foot}} \frac{2.54\; cm}{\cancel{inch}} \frac{30.5\; cm}{\cancel{foot}} = 155\; cm^2

Alternate approach:

By multiplying equations, we can get the unit conversion equation between inch-foot and square centimeters.

1 inch = 2.54 cm 1 foot = 30.48 cm 1\; inch\; foot = 77.4 cm^2

We then multiply by 2 to get the conversion.

2 \cdot 1\; inch\; foot = 77.4 cm^2 \cdot 2

2 \; inch\; foot = 155 cm^2

Converting volumes

Here we show two methods for converting a cubic meter to cubic centimeters.

1 m^3 \cdot \frac{100cm}{m} \cdot \frac{100cm}{m} \cdot \frac{100cm}{m} = 10^6 cm^3 1 m^3 \cdot \left( \frac{100cm}{m} \right)^3 = 1m^3 \cdot \frac{10^6 cm^3}{m^3} = 10^6 cm^3

Alternate approach

Start with the equation

1 meter = 100 cm

multiply the equation by itself three times (cube the equation)

1^3 meter^3 = 100^3 cm^3

1 meter^3 = 1,000,000 cm^3

1 meter^3 = 1 \cdot 10^6 cm^3

We can then use this to convert a quantity:

10 \cdot 1 meter^3 = 1 \cdot 10^6 cm^3 \cdot 10

10 meter^3 = 10^7 cm^3

Converting volumes

How many cubic inches in a cubic foot?

You can use a unit conversion from

inches to feet
from cubic inches to cubic feet

Converting volumes

Convert cubic meters to gallons

Gallons are not based on a length, so you convert directly.