11.4 Arc length and sector area

Arc length and sector area are key concepts in circular geometry. They help us measure parts of circles based on central angles. These calculations are crucial for understanding how circles work and solving real-world problems involving circular shapes.

Formulas for arc length and sector area use the ratio of the central angle to a full circle. This relationship allows us to find missing values like angles or radii. We can even calculate segment areas by combining sector and triangle areas.

Arc Length and Sector Area

Arc length calculation

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• Portion of the circumference of a circle determined by the central angle measure and radius
• Uses formula $Arc\ length = \frac{central\ angle}{360^\circ} \cdot 2\pi r$ where
  • $r$ represents the radius of the circle
  • $central\ angle$ measured in degrees (°)
• Alternative formula $Arc\ length = central\ angle \cdot r$ used when central angle given in radians
• Examples
  • 90° central angle on circle with radius 5 cm results in arc length $\frac{1}{4}$ of circumference ($\frac{1}{4} \cdot 2\pi \cdot 5 = \frac{5\pi}{2}$ cm)
  • $\frac{\pi}{3}$ radian central angle on circle with radius 6 m yields arc length of 2$\pi$ m ($\frac{\pi}{3} \cdot 6 = 2\pi$ m)

Sector area determination

• Region of a circle bounded by two radii and an arc
• Calculated using formula $Sector\ area = \frac{central\ angle}{360^\circ} \cdot \pi r^2$ where
  • $r$ represents the radius of the circle
  • $central\ angle$ measured in degrees (°)
• Alternative formula $Sector\ area = \frac{1}{2} \cdot central\ angle \cdot r^2$ used when central angle given in radians
• Examples
  • 60° central angle on circle with radius 10 in results in sector area $\frac{1}{6}$ of total circle area ($\frac{1}{6} \cdot \pi \cdot 10^2 = \frac{50\pi}{3}$ in$^2$)
  • $\frac{\pi}{4}$ radian central angle on circle with radius 8 cm yields sector area of 16$\pi$ cm$^2$ ($\frac{1}{2} \cdot \frac{\pi}{4} \cdot 8^2 = 16\pi$ cm$^2$)

Central angle from arc length

• Rearrange arc length formula to solve for central angle
  • $central\ angle = \frac{arc\ length}{2\pi r} \cdot 360^\circ$ where
    • $arc\ length$ represents the given length of the arc
    • $r$ represents the radius of the circle
• Alternative formula $central\ angle = \frac{arc\ length}{r}$ used when answer required in radians
• Examples
  • Arc length 6 cm on circle with radius 3 cm results in central angle of 120° ($\frac{6}{2\pi \cdot 3} \cdot 360^\circ = 120^\circ$)
  • Arc length $\frac{\pi}{2}$ m on circle with radius 2 m yields central angle of $\frac{\pi}{4}$ radians ($\frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$ radians)

Segment area by subtraction

• Region of a circle bounded by a chord and an arc
• Calculated by subtracting triangle area from sector area
  1. Calculate sector area containing the segment
  2. Find triangle area formed by two radii and chord using formula $Triangle\ area = \frac{1}{2} \cdot base \cdot height$
  3. Subtract triangle area from sector area to obtain segment area
• Examples
  • Segment formed by 45° central angle on circle with radius 12 ft
    1. Sector area: $\frac{1}{8} \cdot \pi \cdot 12^2 = 18\pi$ ft$^2$
    2. Triangle area: $\frac{1}{2} \cdot 12 \cdot (12 \cdot \sin 45^\circ) \approx 50.91$ ft$^2$
    3. Segment area: $18\pi - 50.91 \approx 5.54$ ft$^2$
  • Segment formed by $\frac{\pi}{6}$ radian central angle on circle with radius 9 m
    1. Sector area: $\frac{1}{2} \cdot \frac{\pi}{6} \cdot 9^2 = \frac{27\pi}{4}$ m$^2$
    2. Triangle area: $\frac{1}{2} \cdot 9 \cdot (9 \cdot \sin \frac{\pi}{6}) \approx 17.56$ m$^2$
    3. Segment area: $\frac{27\pi}{4} - 17.56 \approx 3.69$ m$^2$