An arc and a sector are the fraction of the circle given by the central angle over 360°. Arc length uses the circumference. Sector area uses the area. Circle basics are in circle problems.
The fraction
A 90° central angle is 90/360 = 1/4 of the circle. Arc length is (θ/360) × 2πr. Sector area is (θ/360) × πr². For a radius of 6 and a 60° angle, the fraction is 1/6. The arc is (1/6) × 12π = 2π. The sector is (1/6) × 36π = 6π.
If the angle is given in radians, the fraction of the circle is θ / 2π instead. Do not mix a degree number into the radian formula.
Central, not inscribed
These formulas use the angle at the center. An inscribed angle intercepts an arc twice as large, which is a different question, in inscribed and central angles. Both formulas belong with ACT Math formulas. Line and angle facts outside the circle are in lines and angles.
The bottom line
Find the fraction of 360°, then multiply by circumference for a length or by area for a sector.
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