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Issue No 10, 31 October 2022
By: Anthony O. Ives
The most commonly misunderstood formulae in mathematics are those used to calculate circle circumference and area. Circle circumference is distance around the outside of circle. Circumference is very easy to calculate, it is:
\[C=\pi D=2 \pi R\]
Where C is circumference, D is diameter, and R is radius. π is just a number which is associated with circles which can confuse some people. The value of π is 3.14...., this number is always the same for all circles. If you take different circular objects and measure the diameter which is widest part of the circle and use a piece of string to measure the circumference you can work out π using the rearranged formula:
\[ \pi = \frac{C}{D} \]
This method will probably not give the exact same number for each object measured due to you measuring instrument not being accurate and the possibility the objects are not perfect circle but should be close to 3. The real π could be determined using more accurate methods. Most scientific calculators have π stored in them and can be recalled using a specific function see picture below for example of how to use a scientific calculator to recall π :
Similarly to circumference the equation for the area enclosed in a circle is:
\[ A = \pi R^2 \]
Circles are obviously related to angle so therefore π can also be related to angles. Degrees is the common way to measure angles. To turn a full circle you turn a 360 degrees. If you only want to know the distance of half of the circumference of circle you would divide it by 2, similarly quarter of would be divided by 4 and so on.
If you want a general formula for determining circle geometry as in the diagram above therefore you can assume 1 degree of the circumference can be divided by 360, so a general formula for an arc in terms of degrees is:
\[ S_{arc} = \theta \frac{2 \pi}{360} R \]
In reality angles measured as fractions of π are angles in radians therefore degrees can be converted to radians using this expression:
\[ \theta_{radians} = \frac{2 \pi}{360} \theta_{degrees}= \frac{\pi}{180} \theta_{degrees} \]
Therefore if an angle is converted to radians then the formula for an arc is simply:
\[ S_{arc} = \theta R \]
Similarly the equation for the area enclosed in a circular section is:
\[ A_{arc} = \frac{1}{2} \theta R^2 \]
An understanding of circle geometry is very important for rotorcraft, for example the rotor area needs to be determined using circle formula. The rotor speed at each point on the rotor blade is also determined by calculating circle circumference. The rotor spins a number of revolutions in set time commonly this is given as RPM which is revolutions per minute and really a frequency. So therefore we must divide 60 to give revolutions per second and multiple by it 2π to give angular frequency (can also be referred to as angular speed), the formula would look like this:
\[\omega = RPM \frac{2 \pi}{60} \]
Actual speed which is more specifically known as linear speed is determined by multiplying the circumferential distance of the rotor area by RPM and dividing by 60 to convert to revolution per second, the formula looks like the following:
\[ V= \frac{C}{t}=C \frac{RPM}{60} \]
\[ C =2 \pi R \]
\[ V= 2 \pi R \frac{RPM}{60} \]
As you can see the above formula can simplfied by replacing most of the variables with ω:
\[ V= \omega R \]
This is very interesting as it shows that the speed varies along rotor hence near the hub where the radius is small hence the circumference swept out is also small, it is a low speed. Out towards the tip the radius is large and hence circumference is large and the speed is high. See the diagram for an illustration of this. This makes helicopter more difficult to calculate performance for than a fixed wing aircraft. More informance can be found in Ref [1] and [2] on rotor performance.
Hopefully now you understand circles and π a lot better however if you are still confused by them or more now than ever please let me know by commenting on facebook or via email.
Please also leave a comment on my facebook page or via email and let me know if you understand why circles and π are very important for understanding rotorcraft.
References:
[1] Principles of Helicopter Flight, 2nd Edition, W. J. Wagtendonk, 2006, Aviation Supplies & Academics
[2] Helicopter Theory, Wayne Johnson, 1980, Dover Publications
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