Use radian measure to find arc length, the area of a sector, and the area of a segment of a circle

What this dot point is asking

NESA wants you to use the radian as the natural unit of angle, convert between radians and degrees, and apply the formulas for arc length and sector area, including segments cut off by a chord. Radians are the unit assumed by all calculus involving trig in Maths Advanced.

The answer

Definition of radian

One radian is the angle subtended at the centre of a circle by an arc of length equal to the radius. Equivalently, the radian measure of an angle is the ratio of arc length to radius:

A full revolution is $2 \pi$ radians, because the full circumference $2 \pi r$ divided by the radius $r$ is $2 \pi$.

Conversion

To convert: multiply degrees by $\frac{\pi}{180}$ to get radians; multiply radians by $\frac{180}{\pi}$ to get degrees.

Standard exact values:

Arc length

For a sector of radius $r$ with central angle $\theta$ in radians, the arc length is

This is the formula behind the definition. Use radians, not degrees.

Sector area

The area of a sector of radius $r$ with central angle $\theta$ in radians is

Derivation: the area is the fraction $\frac{\theta}{2 \pi}$ of the full circle area $\pi r^2$, giving $\frac{\theta}{2 \pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta$.

Triangle and segment

The triangle formed by the two radii and the chord has area

The minor segment is the region between the chord and the arc. Its area is

The major segment (the larger region on the other side of the chord) has area $\pi r^2 - A_{\text{segment}}$.

Chord length

By the cosine rule (or by splitting the isosceles triangle), the chord opposite the central angle $\theta$ has length

Worked examples

Converting

$120^\circ = 120 \cdot \frac{\pi}{180} = \frac{2 \pi}{3}$ radians.

$\frac{5 \pi}{4}$ rad $= \frac{5 \pi}{4} \cdot \frac{180}{\pi} = 225^\circ$.

Arc length

A bicycle wheel of radius $35$ cm rotates through $4.5$ radians. Distance travelled by a point on the rim:

$\ell = 35 \cdot 4.5 = 157.5$ cm.

Sector area with angle in degrees

A sector has radius $8$ cm and central angle $45^\circ$. Convert: $45^\circ = \frac{\pi}{4}$ rad.

$A = \frac{1}{2} \cdot 64 \cdot \frac{\pi}{4} = 8 \pi \approx 25.13$ cm$^2$.

Segment area

Circle of radius $5$ cm, central angle $\frac{2 \pi}{3}$.

Sector: $\frac{1}{2} \cdot 25 \cdot \frac{2 \pi}{3} = \frac{25 \pi}{3} \approx 26.18$ cm$^2$.

Triangle: $\frac{1}{2} \cdot 25 \cdot \sin\frac{2 \pi}{3} = \frac{25}{2} \cdot \frac{\sqrt{3}}{2} = \frac{25 \sqrt{3}}{4} \approx 10.83$ cm$^2$.

Segment: $\frac{25 \pi}{3} - \frac{25 \sqrt{3}}{4} \approx 26.18 - 10.83 \approx 15.35$ cm$^2$.

Chord

For $r = 7$, $\theta = \frac{\pi}{3}$:

$c = 2 \cdot 7 \cdot \sin\frac{\pi}{6} = 14 \cdot \frac{1}{2} = 7$ cm. (Consistent with an equilateral triangle: $\frac{\pi}{3}$ at the centre with $r = 7$ gives chord equal to radius.)

Common traps

Using degrees in radian formulas. $\ell = r \theta$ and $A = \frac{1}{2} r^2 \theta$ require $\theta$ in radians. Substituting $90$ instead of $\frac{\pi}{2}$ gives a wildly wrong answer.

Forgetting to subtract the triangle. A segment is not the sector; subtract the triangle.

Wrong formula for the triangle. The triangle area is $\frac{1}{2} r^2 \sin \theta$ (with $\sin \theta$, not $\theta$). Confusing this with the sector formula gives a wrong segment.

Calculator in the wrong mode. Always check that your calculator is in radian mode when working from $\frac{\pi}{6}$ etc.

Confusing the minor and major segment. "Minor" is the smaller piece, between the chord and the shorter arc. Make sure $\theta$ refers to the central angle of that smaller piece (less than $\pi$).

In one sentence

In radians, arc length is $\ell = r \theta$, sector area is $\frac{1}{2} r^2 \theta$, and the segment cut off by a chord has area $\frac{1}{2} r^2 (\theta - \sin \theta)$.