Arc Length, Sector Area and Radian Measure

All circles are similar

Take any two circles, with centers $O$ and $P$ and radii $r$ and $R$.

1. Translate the first circle so $O$ lands on $P$.
2. Dilate about $P$ by the scale factor $\tfrac{R}{r}$.

Every point of the first circle was at distance $r$ from the center; after the dilation it is at distance $r \times \tfrac{R}{r} = R$. Every point at distance $R$ from $P$ is on the second circle, so the first circle lands exactly on the second.

A translation followed by a dilation is a similarity transformation, so any two circles are similar. That is the fact the rest of this lesson is built on.

Why an arc grows with its radius

Draw two circles with the same center and one angle at that center.

The dilation that carries the inner circle to the outer one doubles every length, and a dilation about the center keeps every angle at the center. So the arc cut off on the outer circle is exactly twice the arc on the inner circle.

For a fixed central angle, arc length is proportional to radius:

$$\frac{s_1}{r_1} = \frac{s_2}{r_2}$$

The ratio $\tfrac{s}{r}$ depends only on the angle — not on which circle it was measured on. A quantity that belongs to the angle alone can serve as a measure of the angle, and that is what a radian is.

The radian

$$\theta \text{ (in radians)} = \frac{\text{arc length}}{\text{radius}} = \frac{s}{r}$$

One radian is the angle whose arc is exactly one radius long. On the unit circle, where $r = 1$, the radian measure of an angle is exactly the length of the arc it cuts off.

A full turn cuts off the whole circumference, $2\pi r$:

$$\frac{2\pi r}{r} = 2\pi \text{ radians} = 360°$$

That single equation gives every conversion.

Degrees	Radians
$360°$	$2\pi$
$180°$	$\pi$
$90°$	$\tfrac{\pi}{2}$
$60°$	$\tfrac{\pi}{3}$
$45°$	$\tfrac{\pi}{4}$
$30°$	$\tfrac{\pi}{6}$

$$\text{radians} = \text{degrees} \times \frac{\pi}{180} \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}$$

Arc length

The definition of the radian rearranges straight into the arc length formula:

$$s = r\theta \qquad (\theta \text{ in radians})$$

In degrees, the arc is the same fraction of the circumference as the angle is of a full turn:

$$s = \frac{\theta}{360°} \times 2\pi r$$

Find the length of the arc cut off by a $60°$ angle on a circle of radius $9$ cm.

$$s = \frac{60}{360} \times 2\pi(9) = 3\pi \approx 9.42 \text{ cm}$$

In radians, $60° = \tfrac{\pi}{3}$, and $s = 9 \times \tfrac{\pi}{3} = 3\pi$ — the same answer with less arithmetic. That shortcut is the practical reason radians exist.

Sector area

A sector is the slice of a circle between two radii — a slice of pizza.

The sector is the same fraction of the whole area as its angle is of a full turn. This one spans $90°$, a quarter of the circle.

$$A = \frac{\theta}{360°} \times \pi r^2 \qquad\text{or}\qquad A = \frac{1}{2} r^2 \theta \;\; (\theta \text{ in radians})$$

The radian form comes from the degree form: replace $\tfrac{\theta}{360°}$ with $\tfrac{\theta}{2\pi}$, and $\tfrac{\theta}{2\pi} \times \pi r^2 = \tfrac{1}{2}r^2\theta$.

Why radians are the natural unit

Degrees split a turn into $360$ parts, a number chosen long ago because it divides evenly by so many others. Nothing about a circle makes $360$ special.

A radian is measured by the circle itself — arc against radius, one length against another. That is why the formulas lose their conversion factors: $s = r\theta$ has no $360$ and no $2\pi$ in it.

The same simplicity is why trigonometry beyond right triangles, and all of calculus, is done in radians. A calculator that gives nonsense for $\sin 30$ is usually in radian mode, reading the $30$ as thirty radians.

Worked examples

Common mistakes

Practice problems