There’s an infamous quote attributed to NBA coach and former player Jason Kidd: “We’re going to turn this team around 360 degrees!”
Whoops. Math teachers must’ve rolled their eyes at that one.
As you might already know, a circle is made up of exactly 360 degrees. If Kidd had made good on his promise, then his team would have turned around all right. Trouble is, the squad wouldn’t stop turning until it had “gone full circle” and ended up right back where it started.
Not a recipe for improvement. What Kidd was looking for was to turn his team around 180 degrees and make them winners!
The degree, in this context, is a unit we can use to measure angles. On paper, degrees are represented by the degree symbol, which looks like this: °
So instead of writing “18 degrees,” you could simply write “18°.“
One of the most important concepts in trigonometry and geometry is the right angle. This is the angle that’s formed where two perpendicular lines intersect.
It also represents one quarter of a full rotation.
Let’s say you want to physically turn something. Anything. You’ve chosen a fixed center point and are trying to maneuver that object around it in a circular motion. If you finish the job and make a complete circle, that’s a full rotation. But if you stop the process 25 percent of the way through, that’s only one quarter of a full rotation. Which gives you a right angle.
A right angle is equal to 90 degrees (i.e., 25 percent of 360). Here’s another way of putting it: A right angle is equal to π/2 radians.
Time-out! What’s a radian? And how did π (pi) get mixed up in this?
OK, imagine a perfect circle. There’s a straight line which begins at its exact center and ends at the circle’s curved perimeter (as in, the outer edge).
By definition, that line is the radius of our circle.
Essentially, a radian is a slice of a circle. Look at the circle’s curved outer perimeter one more time. Now imagine a segment of the perimeter that is equal in length to the radius of your circle. If you drew two straight lines connecting its two endpoints to the circle’s exact center, the angle they’d produce would be a radian.
Every circle has room for the same number of radians. That number is equal to 2 times pi (“π”). Since pi itself is equal to about 3.14, you could say there are approximately 6.28 radians in a circle (2 x 3.14). Or that 1 radian is about 57.29 degrees (180°/π).
Radians Vs. Degrees
Look, we won’t deny it. Radians can be a harder concept to visualize than degrees are.
But don’t discount the former. Both of these angle-measuring units have their advantages.
The degree is way more popular. Out in the real world, you’re more likely to encounter people who think in terms of degrees as opposed to radians. So, if you’re trying to communicate with a non-mathematician, maybe stick to degrees.
However, in calculus, radians are great because they lend themselves to much simpler equations. Future A.P. students will want to keep that in mind.
Converting Degrees to Radians (and Vice Versa)
To convert degrees into radians, you just need to memorize a few easy steps.
First, take the number of degrees you wish to convert. Multiply this number by π radians/180 degrees. By eliminating some redundant units and then simplifying things a bit, you’ll have your answer.
Suppose you’ve got a metal bar that’s been bent at a 120 degree angle. How can we express this in terms of radians?
To find out, we’ll write our equation like so:
120° x (π radians/180°)
Notice the pair of degree symbols shown above. Those will cancel each other out, ensuring our final answer will be in radians. We are now left with:
120 x (π radians/180)
Do the multiplication and you get 120π/180 radians. But we’re not quite done yet. Now we’ve got to simplify our fraction if possible. We need to identify the highest whole number that can be divided exactly into both the denominator (180) and the non-π portion of the numerator (120). Spoiler alert: In our case, the magic number is 60.
If you actually divide 120π and 180 by 60, you get 2π/3 radians.
So, there we go: 120° is equal to 2π/3 radians.
Going from radians to degrees is a similar procedure. Only in this case, we’d take the starting amount of radians and multiply it by (180°/π).
π/3 radians x (180°/π) = 60 degrees
To convert from radians to degrees: multiply by 180, divide by π
To convert from degrees to radians: multiply by π, divide by 180