Degree sectors and arc lengths

A central angle selects a fraction of a circle

A sector of angle θ degrees occupies θ/360 of a circle. For radius r, its arc length is (θ/360)(2π r) and its area is (θ/360)(π r²).

For radius 12 cm and angle 75°, arc length is 5π cm and area 30π cm². The complete sector perimeter adds two radii, so it is 24 + 5π cm. Arc length alone leaves out both straight sides.

Major and minor regions use different angles

With the same two radii, the minor sector could be 75° and the major sector 285°. Their arc lengths add to the full circumference and their areas add to the full circle’s area. Read which region is named or shaded before selecting the angle.

For a composite boundary, trace only exposed arcs and straight edges. A shared radius between two joined sectors is interior, so it is not counted in the outside perimeter.

Check your understanding 1 Recover the central angle

A sector of radius 10 cm has arc length 5π cm. Find its central angle in degrees and its area.

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The full circumference is 20π cm, so the arc is one quarter of it. The angle is 90° and area is 25π cm². The arc-to-circumference fraction equals the area-to-circle fraction at the same radius.

A radius change affects area differently

Keep the central angle fixed and double the radius. The arc length doubles, but area quadruples. This follows from the first power of r in circumference and the second power in circle area; it is not a property of the drawing’s apparent size.