Write an expression for the length of an arc whose measure is 𝜃 radians, knowing that the expression for the length of an arc in degrees is two 𝜋𝑟𝜃 over 360.
We recall first that an arc is part of the circumference of a circle. An arc has a central angle, which we often refer to as 𝜃, which is the angle formed by the intersection of the two radii drawn from the endpoints of the arc to the center of the circle. We’re given in the question a formula for calculating an arc length when the central angle of the sector, 𝜃, is measured in degrees. And we want to find a formula we can use when the angle is instead measured in radians. Before we do this, though, let’s remind ourselves where the formula we’ve been given comes from.
We know that the circumference of a full circle is given by the formula 𝜋𝑑 or two 𝜋𝑟, where 𝑟 represents the radius of the circle. When we’re finding an arc length though, we only want to find the length of part of the circumference. The fraction of the circumference that we have is determined by the central angle 𝜃. As there are 360 degrees in a full turn, the portion of the circle we have is 𝜃 over 360. So to find the arc length, we multiply the full circumference of the circle by this fraction, giving two 𝜋𝑟 multiplied by 𝜃 over 360, which, of course, is just two 𝜋𝑟𝜃 over 360 as given in the question.
We can take two approaches to answering this question then. One is to follow the same process from the start but using radians instead of degrees. The other is to convert between degrees and radians in our final formula. Let’s demonstrate both methods. Starting from scratch then, the circumference of a circle as before is two 𝜋𝑟. There are two 𝜋 radians in a full turn. So for a central angle 𝜃 measured in radians, the portion of the circumference that we have is 𝜃 over two 𝜋. Multiplying the full circumference of the circle then by this fraction gives two 𝜋𝑟 multiplied by 𝜃 over two 𝜋.
We see that in fact the two 𝜋 in the numerator will cancel with the two 𝜋 in the denominator. And in this case, the formula for the arc length simplifies to just 𝑟𝜃. We multiply the radius 𝑟 by the central angle 𝜃 measured in radians. The second method, we said, would be to take the formula given in the question for an angle measured in degrees, two 𝜋𝑟𝜃 over 360, and then convert to radians. If we now assume that the angle 𝜃 in the numerator is measured in radians, we can then replace 360 degrees in the denominator with two 𝜋 radians. That gives two 𝜋𝑟𝜃 over two 𝜋. And as before, we see that the factors of two 𝜋 in the numerator and denominator will cancel each other out, leaving just 𝑟𝜃 as before.
So we find that an expression for the length of an arc whose measure is 𝜃 radians is 𝑟𝜃. We multiply the radius by the central angle 𝜃.