Radians and Arc Length

Degrees are a bookkeeping convenience. Splitting the circle into 360 equal parts is convenient for navigation and surveying, but the number $360$ has nothing to do with the geometry of the circle, it is a Babylonian inheritance. Radians are different. They measure angle the way the circle itself wants to be measured: by the length of the arc the angle cuts off, divided by the radius. Once you make that switch, the formulas of calculus, the derivative of sine, the area of a sector, the length of an arc, all snap into place without a stray factor of $\pi/180$.

The definition

Stand at the centre of a circle of radius $r$. Sweep out an arc of length $s$ along the rim. The angle in radians is the ratio

$\theta = \dfrac{s}{r}$

A radian is, literally, "as many radius-lengths of arc as your angle covers." If the arc length equals the radius, the angle is $1$ radian.

Why 2π is a full turn

The circumference of a circle is $2\pi r$, the $\pi$ that already lives inside circle geometry. A full revolution therefore sweeps out $2\pi r$ of arc, which divided by $r$ gives $2\pi$ radians. Half a turn is $\pi$, a quarter is $\pi/2$, and so on. The conversion to degrees follows from $180° = \pi$ rad:

$\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \dfrac{\pi}{180}, \qquad \theta_{\text{deg}} = \theta_{\text{rad}} \cdot \dfrac{180}{\pi}$textdegcdotdfracpi180,qquadthetatextdeg=thetatextradcdotdfrac180pi

The standard angles, commit them to memory

$0,\; \dfrac{\pi}{6},\; \dfrac{\pi}{4},\; \dfrac{\pi}{3},\; \dfrac{\pi}{2}, \; \dfrac{2\pi}{3},\; \dfrac{3\pi}{4},\; \pi,\; \dfrac{3\pi}{2},\; 2\pi$. In degrees: $0°, 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, 360°$. These appear so often in trigonometry and calculus that not memorising them is like trying to read English without knowing the alphabet.

• Astronomy: Angular sizes of stars and galaxies are reported in radians (or arcseconds, where 1 arcsec = $\pi / 648000$ radians); using radians keeps the small-angle approximation $\sin \theta \approx \theta$ clean.
• Robotics: Joint angles in a robot arm are stored in radians because the arc-length $s = r\theta$ formula is immediate; computing how far a finger sweeps when a wrist rotates is just $r \cdot \theta$ with no degrees-to-radians overhead.
• Physics: Angular velocity $\omega$ is in radians per second precisely because then $v = r\omega$ is correct without a unit-conversion factor; degrees would inject a needless $\pi/180$ into every dynamics equation.

(Drag the orange point around the unit circle. Press the snap buttons to jump to standard angles. The readout shows the radian value as a fraction of $\pi$ and as a decimal angle in degrees.)

Arc length and sector area

Once $\theta$ is in radians, the algebra is trivial. Arc length: $s = r\theta$. Sector area: $A = \tfrac{1}{2} r^2 \theta$. Both formulas come straight from the definition $\theta = s/r$; the sector area is just the area of the full disc $\pi r^2$ scaled by the fraction $\theta/(2\pi)$.

Try it

• Predict first: $\pi/4$ radians equals how many degrees? Snap the widget to $\pi/4$ and verify it reads $45°$.
• Compute the arc length on a circle of radius $10$ subtended by an angle of $\pi/3$ rad. Then check using the formula $s = r\theta$.
• Find the area of a slice of pizza cut at $\pi/4$ rad from a pie of radius $12$ cm.

Pause: if you tried to use the sector-area formula $A = \tfrac{1}{2} r^2 \theta$ with $\theta$ in degrees, would it give the right answer? Try $r = 1$, full-circle angle: degrees give $A = 180$, but the actual area is $\pi \approx 3.14$. The factor of $180/\pi \approx 57.3$ is exactly the mismatch.

A trap to watch for

Calculators and programming languages have a mode setting: RAD or DEG. If your calculator is in DEG mode and you type $\sin(\pi/2)$, it interprets $\pi/2 \approx 1.57$ as degrees and returns $\sin(1.57°) \approx 0.027$, far from the expected $1$. Always check the mode before computing a trigonometric value. In every calculus formula in this textbook, radians are assumed unless the angle is written with a degree symbol.

What you now know

You can measure angles as a ratio of arc to radius, convert freely between radians and degrees, list the standard angles in both units, and compute arc length and sector area from a radian measure. The next section uses the radian to define sine and cosine, not as ratios in a right triangle, but as coordinates on the unit circle. That definition will extend without modification to angles bigger than $\pi/2$, to negative angles, and to angles beyond a full turn.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 11, §1, the canonical motivation-first treatment of radian measure.
• Apostol, T. M. (1967). Calculus, Volume 1 (2nd ed.). Wiley. Chapter 2 connects radian measure to the integral definition of $\pi$.
• Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage. Appendix D reviews angles, arc length, and the standard-angle table.