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 m^{2} 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 m^{2} to cm^{2}.

**Solution:**

In order to solve this problem, you must see that there are TWO sides to the 1.00 m^{2} area and that EACH SIDE must be converted to cm.

1) Consider that 1.00 m^{2} as a square that is 1.00 m on each side. Its area is our 1.00 m^{2}:

1.00 m x 1.00 m = 1.00 m^{2}

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 m^{2}

4) Now, multiply out the 100 x 100 and replace the 1.00 m^{2} with the corresponding cm^{2} value

1.00 x 10^{4}cm^{2}= 1.00 m^{2}

**Example #2:** Consider 1.00 km^{2}. Convert it to μm^{2} (square micrometers).

**Solution:**

1) Consider 1.00 km^{2} as a square:

1.00 km x 1.00 km = 1.00 km^{2}

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

1 km = 10^{9}μm

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 10^{9}μm) x (1.00 x 10^{9}μm) = 1.00 km^{2}

4) Multiply out the left-hand side:

1.00 x 10^{18}μm^{2}= 1.00 km^{2}

**Example #3:** Convert 1.0 mg/cm^{2} to kg/m^{2}

**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 / 10

^{6}mg) <--- 10^{6}is the absolute exponential "distance" from milli- to kilo-1.0 mg times (1 kg / 10

^{6}mg) = 1.0 x 10^{-6}kg

This means we now have 1.0 x 10^{-6} kg/cm^{2} and now we focus on the denominator (and ignore the kg portion).

Think of 1 cm^{2} 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}m^{2}(this is what 1 cm^{2}is in m^{2})

Now, replace the cm^{2} unit:

1.0 x 10^{-6}kg / 10^{-4}m^{2}

and simplify:

1.0 x 10^{-2}kg / m^{2}= 0.010 kg / m^{2}

**Example #4:** Calculate the mass in pounds of a uniform column of water 34.6 ft high having an area of 1.00 in^{2} at its base.

34.6 ft times 12 inch / ft = 415.2 inThe formula for volume of a cylinder is πr

^{2}hsince πr

^{2}= the area, we have this:415.2 inch times 1.00 in

^{2}= 415.2 in^{3}Now, we need to compute the conversion from inch cubed to cm cubed:

1 in

^{3}= 1 in times 1 in times 1 in1 in

^{3}= 2.54 cm times 2.54 cm times 2.54 cm = 16.387 cm^{3}415.2 in

^{3}times 16.387 cm^{3}/ 1 in^{3}= 6804 cm^{3}6804 cm

^{3}times 1.00 g/cm^{3}= 6804 g6804 g times 1 lb / 454 g = 15.0 lbs

**Example #5:** 5 m^{2} = ____ cm^{2}

**Solution:**

m = cm x 10^{2}<--- 1 m = 1 cm x 100 = 100 cmm

^{2}= (cm x 10^{2})^{2}m

^{2}= cm^{2}x 10^{4}5 m

^{2}= 5 x 10^{4}cm^{2}5 m

^{2}= 50000 cm^{2}

**Example #6:** 5 cm^{2} = ____ m^{2}

**Solution:**

cm = m x 10¯^{2}<--- 1 cm = 1 m x 0.01 = 0.01 mcm

^{2}= (m x 10¯^{2})^{2}cm

^{2}= m^{2}x 10¯^{4}5 cm

^{2}= 5 x 10¯^{4}m^{2}5 cm

^{2}= 0.0005 m^{2}

**Example #7:** A house has an area of 195 m^{2}. (a) Convert the area to km^{2}. (b) Convert the area to ft^{2}?

**Solution to (a):**

1) Think of the 195 m^{2} 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 km

^{2}(or 1.95 x 10¯^{4}km^{2})

3) The conversion can also be shown this way:

(1 km) ^{2}195 m ^{2}x ––––––– = 1.95 x 10¯ ^{4}km^{2}(10 ^{3}m)^{2}

**Solution to (b):**

Think of the 195 m^{2} 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 values:

(639.764 feet) (3.28084167 feet) = 2098.964 ft^{2}Three sig figs would be the most reasonable answer, so 2.10 x 10

^{3}ft^{2}.This is because 2100 has two SF (not enough) and 2100. has four SF (too many).

**Example #8:** 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 m^{2}

2) Convert square meters to square cm:

(4 m^{2}) (100 cm / 1 m)^{2}= 40000 cm^{2}

3) Written another way:

(100 cm) ^{2}4 m ^{2}x –––––––– = 40000 cm ^{2}(1 m) ^{2}

**Example #9:** How many square picometers are there in 8.47 x 10¯^{6} Mm^{2}?

**Solution:**

1) Here's one way:

(10 ^{18}pm)^{2}8.47 x 10¯ ^{6}Mm^{2}x –––––––– = 8.47 x 10 ^{30}pm^{2}(1 Mm) ^{2}

2) Here's another way (note how I go through meters):

(10 ^{6}m)^{2}(10 ^{12}pm)^{2}8.47 x 10¯ ^{6}Mm^{2}x –––––––– x –––––––– = 8.47 x 10 ^{30}pm^{2}(1 Mm) ^{2}(1 m) ^{2}

**Example #10:** How many square picometer are there in 7.71 x 10¯^{6} square nanometer?

**Solution:**

(10 ^{3}pm)^{2}7.71 x 10¯ ^{6}nm^{2}x –––––––– = 7.71 pm ^{2}(1 nm) ^{2}

**Square unit problems**

1) Convert 4.26 x 10^{4} m^{2} to km^{2}

2) Convert 3.20 x 10^{10} fm^{2} to cm^{2}.

3) Convert the answer in number 2 to Mm^{2}