Arc Length and Sectors — SAT Math Explained
An arc is a portion of a circle's circumference. A sector is the 'pie slice' region bounded by two radii and an arc. Both are proportional fractions of the full circle based on the central angle.
The Core Idea
The central angle of an arc or sector is a fraction of the full 360°. That same fraction of the full circumference gives arc length; that fraction of the full area gives sector area.
Step-by-Step: How to Approach Arc Length and Sectors
Identify the central angle θ and the radius r
Write the fraction: θ/360°
Multiply by the full circumference (2πr) for arc length, or by full area (πr²) for sector area
Simplify and compute
Formulas
L = (θ/360°) × 2πr
A = (θ/360°) × πr²
Proportional Reasoning
If the central angle is 90° (a quarter circle), the arc length is 1/4 of the circumference, and the sector area is 1/4 of the total area.
Radians
In higher math, angles are measured in radians. Arc length = rθ (in radians), making the formula even cleaner.
Common Errors to Avoid
Using diameter instead of radius
Not multiplying by 2π for arc length (accidentally using πr² fraction)
Confusing arc (curved boundary) with sector (filled region)
Practice: Arc Length and Sectors
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Arc length fraction equals:
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