 Suppose we have a central angle theta inside of a circle that cuts off some proportion of this circumference. We see some arc right here. Let's say the length of that arc is equal to s. It is a circles which specify the radius. The radius here, we'll call it r, and then the radian measure of this angle, we're going to call that theta. What we learned previously about radians is that radians, it's a measurement of the angle, yes. But it uses the proportions of the arc length to with the circles radius to determine the measure here. That is to say, when it comes to radian measure, we see that the arc length divided by the radian measure is always going to equal the radius of the circle. If you times both sides of the equation by theta, this then gives you the very useful formula s equals r theta, where the length of the arc is going to equal the radius of the circle times the radian measure of the angle. Now it is important that the measure of the angle must be in radians because this proportionality condition is the basis of why we defined radian measure to be what they were. It's kind of like in chemistry, the superior way to measure temperature is to use Kelvin. It's better than Fahrenheit, it's even better than Celsius because zero degrees Kelvin represents absolute zero. That is no vibration of the molecules there. There's a meaning, a quantitative interpretation of that zero degrees there. Similar is true here with radian measure. There's this proportionality condition between arc length and angle measure. And so we use radians to measure the arc of a circle. So imagine we'd want to find the length of the arc cut off by a central angle, which is measured two radians in a circle and the radius is 4.3 inches. So what we're going to see here is that the angle measure was two radians. The radius is 4.3 and then the arc length, what is the arc length going to be? Well, the formula we just used from above, s is going to equal r times theta, r equals 4.3 inches and then we get two radians here. You don't have to measure, excuse me, you don't have to mention the units when it comes to working with radians because if you mention nothing, then that by default means radian measure. And then a simple multiplication here, 4.3 times two is going to equal 8.6 inches and that's how long the arc will be in that situation. Simple formula, just make sure you're using radians. This is my case in point. If we want to find the length of the arc cut by the central angle of 36 degrees in a circle of radius of two feet, all right? So we're going to use our formula s equals r times theta. The radius was given as two feet, but then the angle here was given as 36 degrees, which if we want to switch to radians, we need to remember that one degree, one degree is the same thing as pi over 180 radians. So we have to make sure we switch to radians here, which then adds this extra factor of pi over 180 right here for which we see that 36 goes into 185 times. So we're going to get two feet times pi over five radians and once then putting that together, we're going to get two pi over five feet would be the exact answer, but as we're measuring distance, we might want to approximate this thing. We would get approximately 1.26 feet when we consult our calculator.