Geometry/Circles/Arcs

An arc is a segment of the perimeter of a given circle. The measure of an arc is measured as an angle, this could be in radians or degrees (more on radians later). The exact measure of the arc is determined by the measure of the angle formed when a line is drawn from the center of the circle to each end point. As an example the circle below has an arc cut out of it with a measure of 30 degrees.



Radians:

As I mentioned before an arc can be measured in degrees or radians. A radian is merely a different method for measuring an angle. If we take a unit circle (which has a radius of 1 unit), then if we take an arc with the length equal to 1 unit, and draw line from each endpoint to the center of the circle the angle formed is equal to 1 radian. this concept is displayed below, in this circle an arc has been cut off by an angle of 1 radian, and therefore the length of the arc is equal to $$2\pi$$ because the radius is 1.



From this definition we can say that on the unit circle a single radian is equal to $$ 2 \pi $$ radians because the perimeter of a unit circle is equal to $$2\pi$$. Another useful property of this definition that will be extremely useful to anyone who studies arcs is that the length of an arc is equal to its measure in radians multiplied by the radius of the circle.

Converting to and from radians is a fairly simple process. 2 facts are required to do so, first a circle is equal to 360 degrees, and it is also equal to $$2\pi$$. using these 2 facts we can form the following formula:

$$2\pi = 360, \pi = 180, \frac{\pi}{180} = 1$$, thus 1 degree is equal to $$\frac{\pi}{180}$$ radians. From here we can simply multiply $$\frac{\pi}{180}$$ by the number of degrees to convert to radians. for example if we have 20 degrees and want to convert to radians then we proceed as follows: $$\left(\frac{\pi}{180}\right)20 = \frac{20\pi}{180} = \frac{\pi}{9}$$ radians.

The same sort of argument can be used to show the formula for getting 1 radian.

$$360 = 2\pi, 180 = \pi, \frac{180}{\pi} = 1$$, thus 1 radian is equal to $$\frac{180}{\pi}$$