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Square and Cubic Units

The SI units of area and volume are the square meter
(m^{2}) and the cubic meter (m^{3}). They are
called derived units because they are built up from simpler
units. In the image below the differences between length, area
and volume units are made clear. To measure a length requires
just one dimension, one measurement. To measure area requires two
dimensions: two lengths multiplied together. To measure volume
requires three dimensions: three lengths multiplied together.

To convert from one unit of area or volume to another can be done in two ways. First, the length conversion factor can be used to make a new conversion factor for area or volume. Second, the length conversion factor can be used more than once: twice for area units and three times for volume units.

Here is how to do this, step-by-step:

- Write out the full conversion factor for converting length (for ex., 1 ft = 12 in).
- For area, square both sides of the equation. For volume, cube both sides of the equation.
- Be sure to square or cube the
*unit*in addition to the number.

1 ft = 12 in 1 mi = 5,280 ft (1 ft)^{2}= (12 in)^{2}(1 mi)^{2}= (5,280 ft)^{2}1 ft^{2}= 144 in^{2}1 mi^{2}= 2.78784 × 10^{7}ft^{2}Now that you have a conversion factor, use it: 144 in^{2}2~~ft~~× --------- = 288 in^{2}^{2}1~~ft~~^{2}

1 m = 100 cm 3 ft = 1 yd (1 m)^{3}= (100 cm)^{3}(3 ft)^{3}= (1 yd)^{3}1 m^{3}= 1 × 10^{6}cm^{3}27 ft^{3}= 1 yd^{3}Now that you have a conversion factor, use it: 1 × 10^{6}cm^{3}3.5~~m~~× ----------- = 3.5 × 10^{3}^{6}cm^{3}1~~m~~^{3}

Here is how to do this, step-by-step:

- Write the number to be converted with its unit.
- If the unit is squared then write the conversion factor for length twice, being sure that the units to be cancelled are on the opposite sides of the fraction bar.
- If the unit is cubed then write the conversion factor for length three times, being sure that the units to be cancelled are on the opposite sides of the fraction bar.

Convert cm^{2}to m^{2}: 1 m 1 m 45,000~~cm~~× ———————— × ———————— = 4.5 m^{2}^{2}100~~cm~~100~~cm~~

Convert m^{3}to ft^{3}3.28 ft 3.28 ft 3.28 ft 5.2~~m~~× ——————— × ——————— × ——————— = 183 ft^{3}^{3}1~~m~~1~~m~~1~~m~~

Do your work on a separate piece of paper, being sure to label each problem clearly. To do these problems you may need your conversions chart from the original dimensional analysis packet and your metric units chart.

Create conversion factors between the two units given in each problem.

- mm
^{2}to cm^{2} - m
^{3}to cm^{3} - in
^{3}to cm^{3} - m
^{3}to km^{3} - μm
^{3}to mm^{3}

- m
^{3}to ft^{3} - km
^{2}to mi^{2} - in
^{2}to ft^{2} - m
^{2}to ft^{2} - m
^{2}to yd^{2}

Perform the following conversions, showing all work.

- 1.2 × 10
^{6}cm^{3}to m^{3} - 5 m
^{3}to yd^{3} - 1.40 × 10
^{2}m^{2}to ft^{2} - 150 mm
^{2}to cm^{2} - 2.5 ft
^{2}to in^{2}

- 1.70 × 10
^{2}km^{2}to cm^{2} - 5 km
^{3}to m^{3} - 56 mi
^{2}to km^{2} - 3.45 × 10
^{4}cm^{3}to L - 4.98 × 10
^{3}mm^{3}to gal

- One square mile contains 640 acres. How many square feet are there in an acre?
- A unit commonly used in teaching organic chemistry in the lab is the microliter (μL). Express the volume of this unit using a unit derived from metric length units. Set up a conversion factor so that 1 μL = 1 of the new unit. Hint: Start with the fact that 1 mL is defined as 1 cm
^{3}.