Intuitive Trigonometry/Radians and Arc Length

Radians and Arc Length

Finding the formula for arc length

First, let us examine the formula for arc length. Imagine some arc, and then extend each of the ends. Eventually, you will end up with a circle (see Figure 1). We know that the circumference of a circle is $2\pi r$, which we can imagine as the circumference of a full circle. Every arc length must then be equal to or a fraction of $2\pi r$.

Next, imagine a circle, with an angle in it that opens onto a section of the circle (see Figure 2). As the angle varies, clearly the arc length varies. In fact, if the angle is $360^{\circ }$ it is easy to see that the arc length must be $2\pi r$. Using this, we can see that to find the length of any arc, all we must do is multiply the circumference of a circle (a "full" arc), $2\pi r$ by the angle measure out of $360^{\circ }$, leaving us with the formula $\ell _{\text{arc}}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)$.

Figure 1: A circle can be thought of as a "full" arc.

Figure 2: Given an angle $\theta$ that subtends an arc on a circle with a radius $r$, one can determine the length of the arc.

Definition of a radian

To derive the conversion rate between radians and degrees, we must know what exactly a radian is. A radian is defined as

A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.

Deriving the conversion rate between radians and degrees

You can derive the relation $2\pi {\text{ rad}}=360^{\circ }$ using the formula for arc length that we just derived. Take the formula for arc length, or $\ell _{arc}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)$. Assume a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that $1=2\pi \left({\frac {1{\text{ rad}}}{360^{\circ }}}\right)$. We can further simplify this to $1={\frac {2\pi {\text{ rad}}}{360^{\circ }}}$. Now we can multiply both sides by $360^{\circ }$ to get $360^{\circ }=2\pi {\text{ rad}}$.

Example conversion between radians and degrees

Suppose we are given the degree measure $30^{\circ }$ and asked to convert it into radians. We know that $2\pi {\text{ rad}}=360^{\circ }$, so we can set up a proportion: ${\frac {x{\text{ rad}}}{30^{\circ }}}={\frac {2\pi {\text{ rad}}}{360^{\circ }}}$. We can then cross multiply, getting $360^{\circ }\times x=2\pi {\text{ rad}}\times 30^{\circ }$. Dividing each side by $360^{\circ }$, notice first that the degree signs cancel, so we are left with $x={\frac {60\pi {\text{ rad}}}{360}}$ which simplifies to $x={\frac {\pi }{6}}{\text{ rad}}$. The problem is solved (notice it is customary to leave it in fraction form as it is more exact).

Problems

All solutions in Appendix 1.

1. Find the length of an arc whose related angle is $10^{\circ }$ and whose related circle has a radius of $2$.
2. Convert ${\frac {\pi }{2}}{\text{ rad}}$ into degrees.
3. Convert $45^{\circ }$ into radians.
4. Find the measure of the angle related to an arc of length $5$ on a circle with a radius of length $4$.