What is the equivalent measure of 360 degrees in radians?
We know that an angle of 360 degrees represents one full revolution. So the question then becomes “what is the measure of one full revolution in radians?” To do this, we use the definition of the measure of an angle in radians. It’s equal to the length of the intercepted arc divided by the radius of the circle. So this 𝜃 that we’re trying to find the measure of is the angle in the diagram. And we want to apply our formula to this angle.
Let’s start with the radius. We’re not told what the radius is in the question. And so we assume it can take any value. And so we can call this value 𝑟. So now we’ve decided what to call the radius, we can move on to the arc length. This is the length of the arc intercepted by the angle.
And as the angle is one full revolution, the arc intercepted is the entire circumference of the circle. And we know what the circumference of the circle is in terms of our radius 𝑟; it’s just two 𝜋𝑟.
Now that we found that the arc length is two 𝜋𝑟 and the radius is 𝑟, we can simplify. We cancel the 𝑟s and we’re left with just two 𝜋. So the equivalent measure of 360 degrees in radians is two 𝜋 radians.