# Trig circle showing relationship between degrees and radians Degrees (Trigonometry) The feedback you provide will help us show you more relevant content in the future. Undo Radian is a measure of an angle, indicating the relationship between the arc length and the radius of a circle: Since the arc length of a full circle is the same as the circumference of a circle given by. There are two units that we can use to measure angles: degrees and radians. In this lesson The unit circle is a circle centered at the origin (0,0) that has a radius of exactly one. It is used to relationship between angle and arc length Common angles (measured in both radians and degrees) shown on the unit circle. Think about what the word radian sounds like well, it sounds like 'radius', right? It turns out that a radian has a close relationship to the radius of a circle.

Instead of wondering how far we tilted our heads, consider how far the other person moved. Degrees measure angles by how far we tilted our heads.

Radians measure angles by distance traveled. So we divide by radius to get a normalized angle: Moving 1 radian unit is a perfectly normal distance to travel. Strictly speaking, radians are just a number like 1. Now divide by the distance to the satellite and you get the orbital speed in radians per hour.

Sine, that wonderful function, is defined in terms of radians as This formula only works when x is in radians! Well, sine is fundamentally related to distance moved, not head-tilting. Ok, the wheels are going degrees per second. Now imagine a car with wheels of radius 2 meters also a monster.

Wow -- the car was easier to figure out than the bus! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover. The reverse is easy too. How fast are the wheels turning? Calculus is about many thingsand one is what happens when numbers get really big or really small. Choose a number of degrees xand put sin x into your calculator: When you make x small, like. Even stranger, what does it mean to multiply or divide by a degree? Can you have square or cubic degrees? Radians to the rescue! If you go an even smaller amount, from 0 to. We just have to remember, when we're measuring in terms of radians, we're really talking about the arc that subtends that angle. So if you go all the way around, you're really talking about the arc length of the entire circle, or essentially the circumference of the circle. And you're essentially saying, how many radius's this is, or radii, or how many radii is the circumference of the circle. You know a circumference of a circle is two pi times the radius, or you could say that the length of the circumference of the circle is two pi radii.

If you wanna know the exact length, you just have to get the length of the radius and multiply it by two pi. That just comes from the, really, actually the definition of pi, but it comes from what we know as the formula for the circumference of a circle. If we were to go all the way around this, this is also two pi radians. That tells us that two pi radians, as an angle measure, is the exact same thing, and I'm gonna write it out, as degrees. And then we can take all of this relationship and manipulate it in different ways. If we wanna simplify a little bit, we can divide both sides of this equation by two, in which case, you are left with, if you divide both sides by two, you are left with pi radians is equal to degrees.

## Intuitive Guide to Angles, Degrees and Radians

How can we use this relationship now to figure out what degrees is? Well, this relationship, we could write it in different ways. We could divide both sides by degrees, and we could get pi radians over degrees is equal to one, which is just another way of saying that there are pi radians for every degrees, or you could say, pi over radians per degree.

The other option, you could divide both sides of this by pi radians. You could say, you would get on the left hand side you'd get one, and you would also get, on the right hand side, you would get degrees for every pi radians. Or you could interpret this as over pi degrees per radian. How would we figure out, how would we do what they asked us?

Let's convert degrees to radians.

### Intro to radians (video) | Trigonometry | Khan Academy

Let me write the word out. Well, we wanna convert this to radians, so we really care about how many radians there are per degree, actually, let me do that in that color.

We'll do that same green color. How many radians are there per degree? Well, we already know, there's pi radians for every degrees, or there are pi Let me do that yellow color.

There are pi over radians per degree. And so, if we multiply, and this all works out because you have degrees in the numerator, degrees in the denominator, these cancel out, and so you are left with times pi divided by radians. So what do we get?