A sector is a fraction of a circle
A central angle theta selects theta/360 of both the circumference and the circular area.
Learning goals
- Distinguish arc, sector and central angle.
- Select and apply arc-length and sector-area formulae.
- Recover an unknown radius or angle from sector data.
Concept lesson
Whole-circle measures
For radius r, circumference is 2 pi r and area is pi r squared. If diameter is given, first use r=d/2.
Central-angle fraction
A sector of angle theta occupies theta/360 of the full circle; 180 degrees is one half and 90 degrees one quarter.
Arc, sector and central angle
An arc is a curved boundary; a sector is the region enclosed by two radii and that arc; theta is measured at the centre.
Two proportional formulae
Arc length=(theta/360)(2 pi r), while sector area=(theta/360)(pi r squared). Length uses linear radius; area uses radius squared.
Arc and sector from one angle
For r=21 cm and theta=60 degrees, use one-sixth of 2 pi r for arc length and one-sixth of pi r squared for sector area.
Sector perimeter
Perimeter of a sector equals its arc length plus two radii. Include only the boundary requested by the diagram.
Clock sweep conversion
A minute hand turns 360 degrees in 60 minutes, so t minutes corresponds to 6t degrees; use that angle for swept area or arc length.
Choose by units and target
Classify the target as length or area, normalize diameter to radius, select the matching whole-circle measure and multiply by theta/360.
Swept area and boundary
Convert time or equal divisions to an angle, then separately compute swept area, arc length and full sector perimeter when requested.
Work backwards from sector data
Substitute known arc length, sector area or ratio into the appropriate formula, isolate radius first when both radius and angle are unknown, then recover theta.
Linked sector measures
Use the shared theta/360 factor to connect arc length and sector area, solve unknown measures and verify length and area units independently.
Formula-to-inverse exam ladder
One-mark items test formula recognition; two-mark items test direct substitution; three-mark items link area, arc, radius, angle or perimeter.
Arc and sector mastery
Identify a sector, calculate its arc and area, include radial boundaries where required, and solve one inverse measure problem.