Radian Measure and Arc Length

Degrees are one way to measure an angle, but trigonometry often uses a more natural unit called the radian. A radian is defined using the circle itself: it is the angle formed at the center of a circle when the arc it cuts off is exactly as long as the radius.

Picture a circle of radius \(r\). Walk a distance of \(r\) along the circle's edge starting from a point, then draw a line from your starting point and your ending point back to the center. The angle between those two radius lines is 1 radian, no matter how big or small the circle is. This is why radians connect so cleanly to arc length, unlike degrees, which are just an arbitrary split of a circle into 360 equal parts.

Since one full trip around a circle is 360 degrees, and that same trip corresponds to an angle of \(2\pi\) radians, the two units are linked by the key relationship:

\(\pi \) radians\( = 180^\circ\)

From this, you get two conversion rules:

To convert degrees to radians: \(\theta_{radians} = \theta_{degrees} \times \dfrac{\pi}{180}\)

To convert radians to degrees: \(\theta_{degrees} = \theta_{radians} \times \dfrac{180}{\pi}\)

Example 1: Convert \(45^\circ\) to radians.

\(45 \times \dfrac{\pi}{180} = \dfrac{\pi}{4}\)

Example 2: Convert \(\dfrac{5\pi}{6}\) radians to degrees.

\(\dfrac{5\pi}{6} \times \dfrac{180}{\pi} = 150^\circ\)

These conversions come up constantly once you start locating angles on the unit circle, including when you work out a reference angle or apply the ASTC rule to determine the sign of a trig ratio in a given quadrant.

Once an angle is written in radians, finding the length of the arc it cuts off is direct. For a circle of radius \(r\) and a central angle \(\theta\) measured in radians:

\(s = r\theta\)

This formula only works when \(\theta\) is in radians. If you are given an angle in degrees, convert it to radians first before multiplying by \(r\).

It is worth checking that this matches what you already know from the circumference of a circle: a full revolution is \(\theta = 2\pi\), so \(s = r(2\pi) = 2\pi r\), exactly the circumference formula. The arc length formula is really just a generalization of circumference to any fraction of a circle, building on what you learned about arcs of a circle.

Example 3: A circle has radius \(10\) cm. Find the length of the arc cut off by a central angle of \(\dfrac{2\pi}{3}\) radians.

\(s = r\theta = 10 \times \dfrac{2\pi}{3} = \dfrac{20\pi}{3} \approx 20.94 \) cm\(\)

Example 4: An arc has length \(15\) cm on a circle of radius \(6\) cm. Find the central angle in radians, then in degrees.

Rearranging \(s = r\theta\) gives \(\theta = \dfrac{s}{r}\):

\(\theta = \dfrac{15}{6} = 2.5 \) radians\(\)

Converting to degrees: \(2.5 \times \dfrac{180}{\pi} \approx 143.2^\circ\)

Because \(s = r\theta\) is linear in \(\theta\) for a fixed radius, the arc length grows steadily as the central angle increases. The graph below shows arc length \(s\) against \(\theta\) (in radians) for a circle with radius \(r = 5\).

The most frequent error is plugging an angle in degrees straight into \(s = r\theta\). The formula only produces the correct arc length when \(\theta\) is in radians, so always convert first if an angle is given in degrees. Another common slip is mixing up which conversion factor to use; a quick way to check is to remember that radian measures of common angles are usually written with \(\pi\) in them (like \(\dfrac{\pi}{4}\) or \(\dfrac{5\pi}{6}\)), while degree measures are plain numbers like \(45\) or \(150\).

These same radian ideas carry forward into later trigonometry work, such as using the law of cosines or working through applications of the sine law and cosine law, so getting comfortable with radians now pays off later.

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