### Metric Squared Units: Converting from one square unit to another

In solving metric units that involve a square, I believe it is helpful to visualize the square unit to be converted as a square. By this, I mean to visualize something like 1.00 m2 like this: where each side is 1.00 meter in length. To compute the area, we would multiply side by side.

Below, when the conversion of a square unit is carried out, each of the two sides will first be converted and then, multiplied together.

Example #1: Convert 1.00 m2 to cm2.

Solution:

In order to solve this problem, you must see that there are TWO sides to the 1.00 m2 area and that EACH SIDE must be converted to cm.

1) Consider that 1.00 m2 as a square that is 1.00 m on each side. Its area is our 1.00 m2:

1.00 m x 1.00 m = 1.00 m2

2) What we must do next is convert each meter to centimeter:

1.00 m = 100 cm

3) Now, we replace each meter value of our square with the corresponding centimeter value:

100 cm x 100 cm = 1.00 m2

4) Now, multiply out the 100 x 100 and replace the 1.00 m2 with the corresponding cm2 value

1.00 x 104 cm2 = 1.00 m2

Example #2: Consider 1.00 km2. Convert it to μm2 (square micrometers).

Solution:

1) Consider 1.00 km2 as a square:

1.00 km x 1.00 km = 1.00 km2

2) Convert kilo- to its equivalent in units of micro-:

1 km = 109μm

If you do not know how (or remember how) to convert metric units, you need to learn that techniqe to do the above conversion.

3) Replace km with its equivalent μm measurement:

(1.00 x 109μm) x (1.00 x 109μm) = 1.00 km2

4) Multiply out the left-hand side:

1.00 x 1018μm2 = 1.00 km2

Example #3: Convert 1.0 mg/cm2 to kg/m2

Solution:

I will convert first the numerator, then the denominator.

In the numerator, you must convert from mg to kg, like this:

1.0 mg times (___ kg / ___mg)

1.0 mg times (1 kg / ___mg) <--- I always put '1' associated with the larger unit, kilo- in this case.

1.0 mg times (1 kg / 106 mg) <--- 106 is the absolute exponential "distance" from milli- to kilo-

1.0 mg times (1 kg / 106 mg) = 1.0 x 10-6 kg

This means we now have 1.0 x 10-6 kg/cm2 and now we focus on the denominator (and ignore the kg portion).

Think of 1 cm2 as a square like this:

1 cm by 1 cm

What we need to do is convert cm to m, a fairly easy conversion:

1 cm = 0.01 m

Let's replace cm with m:

0.01 m x 0.01 m = 10-4 m2 (this is what 1 cm2 is in m2)

Now, replace the cm2 unit:

1.0 x 10-6 kg / 10-4 m2

and simplify:

1.0 x 10-2 kg / m2 = 0.010 kg / m2

Example #4: Calculate the mass in pounds of a uniform column of water 34.6 ft high having an area of 1.00 in2 at its base.

34.6 ft times 12 inch / ft = 415.2 in

The formula for volume of a cylinder is πr2h

since πr2 = the area, we have this:

415.2 inch times 1.00 in2 = 415.2 in3

Now, we need to compute the conversion from inch cubed to cm cubed:

1 in3 = 1 in times 1 in times 1 in

1 in3 = 2.54 cm times 2.54 cm times 2.54 cm = 16.387 cm3

415.2 in3 times 16.387 cm3 / 1 in3 = 6804 cm3

6804 cm3 times 1.00 g/cm3 = 6804 g

6804 g times 1 lb / 454 g = 15.0 lbs

Example #5: 5 m2 = ____ cm2

Solution:

m = cm x 102 <--- 1 m = 1 cm x 100 = 100 cm

m2 = (cm x 102)2

m2 = cm2 x 104

5 m2 = 5 x 104 cm2

5 m2 = 50000 cm2

Example #6: 5 cm2 = ____ m2

Solution:

cm = m x 10¯2 <--- 1 cm = 1 m x 0.01 = 0.01 m

cm2 = (m x 10¯2)2

cm2 = m2 x 10¯4

5 cm2 = 5 x 10¯4 m2

5 cm2 = 0.0005 m2

Example #7: A house has an area of 195 m2. (a) Convert the area to km2. (b) Convert the area to ft2?

Solution to (a):

1) Think of the 195 m2 as a rectangle 195 m by 1 m.

2) Now, convert each measurement to km:

[(195 m) (1 km / 1000 m)] x [(1 m) (1 km / 1000 m)]

0.195 m x 0.001 m = 0.000195 km2 (or 1.95 x 10¯4 km2)

3) The conversion can also be shown this way:

 (1 km)2 195 m2 x ––––––– = 1.95 x 10¯4 km2 (103 m)2

Solution to (b):

Think of the 195 m2 as a rectangle 195 m by 1 m as you read the following.

I'm going to convert each metric amount to feet, but I first need to do a bit of explanation.

I have memorized the following two conversions:

1 m = 39.3701 inches
12 inches = 1 foot

First, I'm going to convert 195 m:

(195 m) (39.3701 inch / m) (1 foot / 12 inch) = 639.764 feet

Second, convert the 1 m value:

 39.3701 inch 1 foot 1 m x ––––––––––– x –––––– = 3.28084167 feet 1 m 12 inch

Last, multiply the two vales:

(639.764 feet) (3.28084167 feet) = 2098.964 ft2

Three sig figs would be the most reasonable answer, so 2.10 x 103 ft2.

This is because 2100 has two SF (not enough) and 2100. has four SF (too many).

Example #: How many 1 cm squares would it take to construct a square that is 2 m on each side?

Solution:

1) The area in meters squared is:

(2 m) (2 m) = 4 m2

2) Convert square meters to square cm:

(4 m2) (100 cm / 1 m)2 = 40000 cm2

3) Written another way:

 (100 cm)2 4 m2 x –––––––– = 40000 cm2 (1 m)2

Square unit problems

1) Convert 4.26 x 104 m2 to km2

2) Convert 3.20 x 1010 fm2 to cm2.

3) Convert the answer in number 2 to Mm2