11.4 Arc length and sector area

Arc length and sector area let you measure parts of a circle rather than the whole thing. Both formulas work by finding what fraction of the full circle your central angle represents, then applying that fraction to the circumference or area.

Arc Length and Sector Area

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Arc length calculation

Arc length is simply a portion of a circle's circumference. The central angle tells you what fraction of the full 360° you're dealing with, and you multiply that fraction by the total circumference.

$Arc\ length = \frac{central\ angle}{360°} \cdot 2\pi r$

• $r$ is the radius of the circle
• The central angle is measured in degrees

If the central angle is given in radians instead, the formula simplifies to:

$Arc\ length = central\ angle \cdot r$

Examples:

• A 90° central angle on a circle with radius 5 cm: that's $\frac{90}{360} = \frac{1}{4}$ of the circumference, so the arc length is $\frac{1}{4} \cdot 2\pi \cdot 5 = \frac{5\pi}{2} \approx 7.85$ cm.
• A $\frac{\pi}{3}$ radian central angle on a circle with radius 6 m: arc length is $\frac{\pi}{3} \cdot 6 = 2\pi \approx 6.28$ m.

Sector area determination

A sector is the "pizza slice" region bounded by two radii and the arc between them. The same fraction idea applies, but now you're taking a fraction of the total area instead of the circumference.

$Sector\ area = \frac{central\ angle}{360°} \cdot \pi r^2$

When the central angle is in radians:

$Sector\ area = \frac{1}{2} \cdot central\ angle \cdot r^2$

Examples:

• A 60° central angle on a circle with radius 10 in: that's $\frac{60}{360} = \frac{1}{6}$ of the total area, so the sector area is $\frac{1}{6} \cdot \pi \cdot 10^2 = \frac{100\pi}{6} = \frac{50\pi}{3} \approx 52.36$ in$^2$.
• A $\frac{\pi}{4}$ radian central angle on a circle with radius 8 cm: sector area is $\frac{1}{2} \cdot \frac{\pi}{4} \cdot 8^2 = \frac{1}{2} \cdot \frac{\pi}{4} \cdot 64 = 8\pi \approx 25.13$ cm$^2$.

Central angle from arc length

Sometimes you know the arc length and radius but need to find the central angle. Just rearrange the arc length formula:

$central\ angle = \frac{arc\ length}{2\pi r} \cdot 360°$

For the answer in radians:

$central\ angle = \frac{arc\ length}{r}$

Examples:

• Arc length of 6 cm on a circle with radius 3 cm: $\frac{6}{2\pi \cdot 3} \cdot 360° = \frac{6}{6\pi} \cdot 360° = \frac{360°}{\pi} \approx 114.6°$ Watch the arithmetic here. A common mistake is to cancel the 6's and forget the $\pi$ in the denominator. The fraction $\frac{6}{6\pi}$ simplifies to $\frac{1}{\pi}$, not 1.

• Arc length of $\frac{\pi}{2}$ m on a circle with radius 2 m: $\frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$ radians.

Segment area by subtraction

A segment is the region between a chord and the arc it cuts off. It's not the same as a sector. To find a segment's area, you subtract the triangle formed by the two radii and the chord from the sector that contains it.

Steps:

1. Calculate the sector area using the central angle and radius.
2. Find the area of the triangle formed by the two radii and the chord. For a triangle with two sides equal to $r$ and an included angle $\theta$, use: $Triangle\ area = \frac{1}{2} r^2 \sin\theta$
3. Subtract: $Segment\ area = Sector\ area - Triangle\ area$

The triangle formula $\frac{1}{2} r^2 \sin\theta$ comes from the general formula $\frac{1}{2}ab\sin C$ where both sides $a$ and $b$ equal the radius.

Examples:

• Segment formed by a 45° central angle on a circle with radius 12 ft:

  1. Sector area: $\frac{45}{360} \cdot \pi \cdot 12^2 = \frac{1}{8} \cdot 144\pi = 18\pi \approx 56.55$ ft$^2$
  2. Triangle area: $\frac{1}{2} \cdot 12^2 \cdot \sin 45° = 72 \cdot \frac{\sqrt{2}}{2} = 36\sqrt{2} \approx 50.91$ ft$^2$
  3. Segment area: $18\pi - 36\sqrt{2} \approx 56.55 - 50.91 \approx 5.64$ ft$^2$

• Segment formed by a $\frac{\pi}{6}$ radian (30°) central angle on a circle with radius 9 m:

  1. Sector area: $\frac{1}{2} \cdot \frac{\pi}{6} \cdot 9^2 = \frac{81\pi}{12} = \frac{27\pi}{4} \approx 21.21$ m$^2$
  2. Triangle area: $\frac{1}{2} \cdot 9^2 \cdot \sin\frac{\pi}{6} = \frac{81}{2} \cdot \frac{1}{2} = \frac{81}{4} = 20.25$ m$^2$
  3. Segment area: $\frac{27\pi}{4} - \frac{81}{4} \approx 21.21 - 20.25 \approx 0.96$ m$^2$