Areas Related to Circles
Areas Related to Circles tests circumference, area, sectors, segments, and combinations of plane circular figures. Results often use π = 22/7 or 3.14 as directed.
Core rules
- Circumference = 2πr; area of a circle = πr^2.
- Arc length of a sector with angle θ: (θ/360)×2πr.
- Area of a sector: (θ/360)×πr^2.
- Area of a minor segment = area of sector - area of the triangle formed by the two radii and the chord.
- Area of a ring or annulus = π(R^2-r^2), where R is the outer radius and r the inner radius.
Worked example
Find the area and arc length of a sector of radius 7 cm with central angle 90°. Sector area = (90/360)×(22/7)×7^2 = 1/4×22/7×49 = 38.5 cm^2. Arc length = (90/360)×2×(22/7)×7 = 1/4×44 = 11 cm. The sector is a quarter of the circle, so these values are one-fourth of the full area and circumference.
Traps
Use the angle in degrees with θ/360; radian measures need a different formula. Distinguish sector area from segment area: the segment subtracts the triangle. For a major sector, use 360°-θ. Keep radius and diameter separate, and do not round π too early.