Calculate arc lengths, sector areas and angles

Geometry and measures • Mensuration and calculation

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Practical check for arc length answers.

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Check that arc length < circumference (2πr) and scales proportionally with θ.

Key concepts

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Definitions: arc, sector and segment

An arc denotes part of a circle's circumference between two points. A sector denotes the region bounded by two radii and the included arc; its size depends on the central angle. A segment denotes the region bounded by a chord and the corresponding arc; a segment equals a sector minus the isosceles triangle formed by the two radii.

Arc length in degrees and radians

Arc length uses the proportion of the circle's circumference equal to the central angle. In degrees: arc length = (θ/360) × 2πr, where θ is in degrees and r is the radius. In radians: arc length = rθ, where θ is in radians; this form requires no factor of 2π and avoids extra conversion steps.

Sector area in degrees and radians

Sector area uses the proportion of the circle's area that the central angle represents. In degrees: sector area = (θ/360) × πr^2. In radians: sector area = 1/2 × r^2 × θ, where θ is in radians. The radian formula simplifies many calculations when angles are already in radians.

Finding angle from arc length and vice versa

An angle follows directly from arc length by rearranging arc length formula. In radians: θ = arc length / r. In degrees: θ = (arc length / (2πr)) × 360. Conversely, arc length follows by substituting θ into the arc length formula. Unit consistency determines which rearranged form to use.

Area of a segment

Segment area equals sector area minus the area of the isosceles triangle formed by the two radii. If θ is in radians, triangle area = 1/2 × r^2 × sin θ. Segment area (radians) = 1/2 × r^2 × (θ - sin θ). For degree measures, convert θ to radians before using the sine-based triangle area.

Limits, units and common errors

Angles must lie between 0° and 360° (0 to 2π radians) for standard sectors; major and minor sectors require explicit choice of central angle. The angle unit must match the formula: radian formulae require radians, degree formulae require degrees. Common errors include forgetting conversion, using diameter instead of radius, and misidentifying sector versus segment.

Key notes

Important points to keep in mind

Always use radius, not diameter, in arc and sector formulae.

Ensure angle units match the formula: radians for rθ and 1/2 r^2θ, degrees for θ/360 × 2πr and θ/360 × πr^2.

Convert degrees to radians by multiplying by π/180 before using radian formulae.

Segment area = sector area − triangle area; compute triangle area with 1/2 r^2 sin θ (θ in radians).

Check reasonableness: arc length must be less than circumference, sector area less than whole circle area.

Specify whether a sector is major or minor when the central angle could be >180°.

Use exact forms with π for exact answers; use decimal approximations only when required.

For inverse problems, rearrange formulae first to isolate the unknown, then substitute numbers.