Definition: Arc length (s) is the distance along the arc of a circle, defined by an angle Theta (in radians) and the radius (r).
Formula: (s = \Theta \times r)
Example 1:
Given: (\Theta = 5) radians, (r = 12) T
Calculation: (s = 5 \times 12 = 60) fet
Example 2:
Given: (\Theta = 150°), (r = 8) cm
Conversion: Degrees to radians (\Theta = \frac{5\pi}{6})
Calculation: (s = \frac{5\pi}{6} \times 8 = \frac{40\pi}{6} = \frac{20\pi}{3} \approx 20.944) cm
Arc Length in Degrees
Alternative Formula: (s = \frac{\Theta}{360} \times 2\pi r)
Used when (\Theta) is in degrees.
Sector Area
Definition: The area of a sector is a portion of the circle's area, defined by angle (\Theta) and radius (r).
Formula (Radians): (A = \frac{1}{2} \Theta r^2)
Example:
Given: (\Theta = 2) radians, (r = 8) cm
Calculation: (A = \frac{1}{2} \times 2 \times 8^2 = 64) square cm
Formula (Degrees): (A = \frac{\Theta}{360} \times \pi r^2)
Example:
Given: (\Theta = 90°), (r = 10) cm
Calculation: (A = \frac{90}{360} \times 100\pi = 25\pi) square cm
Key Concepts
Radian and Degree Conversion:
(\Theta) must be in radians for direct formulas with radians; otherwise, convert degrees to radians.
Relation of Arc Length and Sector Area:
Both use a fraction of the circle as determined by (\Theta).
Arc Length: Fraction of the circle's circumference.
Sector Area: Fraction of the circle's total area.
Additional Information
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Other Topics Covered: Linear and angular speed, right triangle trigonometry, and various trigonometric formulas and identities.