This converter allows you to find equivalent values of 2 metric units of area, the square millimeters ($mm^2$) and the square centimeters ($cm^2$).
Start by choosing the preferred spelling, which will affect how the word ‘meter’ is spelled. You can choose the American spelling of ‘meter’ or the British spelling of ‘metre’. This will directly modify the spelling of both of our units, as the core word for them is ‘meter’.
Next, choose your input unit out of the 2 offered options ($cm^2$ or $mm^2$) in the ‘CONVERT FROM’ section.
You make a choice of the output unit out of the same 2 options in the ‘CONVERT TO’ section of the converter.
An alternative way to go about unit selection is to simply stick with the default settings or to swap them around by clicking on the icon with 2 arrows headed in opposite directions.
Once you are happy with the unit selection, type the input value into the ‘VALUE TO CONVERT’ section of the converter. Write this value as a decimal number using the decimal dot.
Choose the number of decimal places you want your result rounded toward and click on ‘CONVERT’ to receive the result of your conversion.
Your result will appear below the converter as a decimal number rounded to the desired number of decimal places.
Additionally, you will also receive a conversion rate between your input and output values, alongside a convenient ‘COPY’ icon, which will allow you to easily copy and paste the result elsewhere.
Conversion between the 2 units of area is defined by their conversion rate. Since both units utilize the word ‘square’, it is clear that we can define them as squares with sides defined as units of length.
Hence, a square millimeter is equivalent in area to a square with a side length of 1 mm. This yields an area of (1 mm) x (1 mm) = 1 $mm^2$.
Similarly, a square centimeter is equivalent in area to a square with a side length of 1 cm. However, for our conversion rate to work, we will define this area in millimeters as well. We know that 1 cm is equivalent to 10 mm. Hence, expressing a square centimeter in square millimeters is a question of performing (10 mm) x (10 mm) = 100 $mm^2$.
From here we can see, that the conversion rate of 1 $cm^2$ to $mm^2$ is 1:100.
We can derive 2 formulae from this conversion rate.
The first formula is more suitable for converting $mm^2$ into $cm^2$, as the output ($cm^2$) is also the subject (the standalone variable on the left side) of the formula.
Alternatively, but for similar reasons, the second formula will be used to convert $cm^2$ into $mm^2$.
The following 2 problems will demonstrate the usage of the 2 formulae in practice.
EXAMPLE 1: A flat computer chip has a surface area of 172 $mm^2$. What is the surface area of this chip in $cm^2$?
Since our input is in $mm^2$ and our output is in $cm^2$, we will be using the first formula for solving this problem. We substitute 172 for $mm^2$ and get the following calculations.
EXAMPLE 2: A paper has a printing area of 312.5 $cm^2$. What is the printing area in square millimeters?
We see that our input is in $cm^2$, while the output is in $mm^2$. This makes the second formula a suitable one for solving this problem. We substitute 312.5 for $cm^2$ and count as follows.
As was shown in the previous section, the conversion rate between $mm^2$ and $cm^2$ is defined as 100.
As we know, multiplying and dividing by 100 is a fairly easy task, as we can use the following rules in order to do it from memory.
Let’s have a look at 2 examples that demonstrate this.
EXAMPLE 1: Convert 7.2 $cm^2$ to $mm^2$.
We must multiply the value by 100, hence we move the decimal dot 2 positions to the right. There is only one digit after the decimal dot available, hence we must add a zero at the end. The solution is 720 $mm^2$.
EXAMPLE 2: Convert 1.9 $mm^2$ to $cm^2$.
We must divide the value by 100, hence we move the decimal dot 2 positions to the left. There is only one digit before the decimal dot available, hence we must add a zero in front of it, alongside an extra zero before the decimal dot. This results in 0.019 $cm^2$.
A square millimeter is an incredibly small unit of area, representing a hundredth of a square centimeter, or a millionth of a square meter.
That is why it is not so often we see expressions using this unit in everyday life.
One of the areas where this is used is the size of various gauges. Gauges are an engineering term that can refer to a variety of things, but in this context, it will refer to the thickness of common wires, alongside their ability to transfer energy. We will provide a few examples below of commonly used wire sizes. The area in $mm^2$ refers to the area a circle created by a cross-section of the said gauge would have. Keep in mind that AWG stands for American Wire Gauge, and is an arbitrary measurement that is inversely proportional to the thickness of each gauge. This means that the thicker the gauge, the lower the AWG number.
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