Circles Chords Arcs and Sectors: Explained with Examples

A chord is a straight segment joining any two points on a circle. An arc is a curved part of the circumference between two points, and a sector is the pie-slice region bounded by two radii and their arc — like a slice of pizza.

These ideas matter because circles appear in measurement everywhere: wheels, pipes, fields, and pie charts. Chords, arcs, and sectors give you the vocabulary to describe any part of a circle and the formulas to measure it.

Key Concepts

• Chord: a segment with both endpoints on the circle. The diameter is the longest chord; every diameter is a chord, but not every chord is a diameter.
• Arc: a portion of the circumference. The minor arc is the shorter one between two points; the major arc is the longer one. A semicircle is an arc of exactly half the circle.
• Central angle: the angle at the center cutting off an arc or sector, measured in degrees (full circle = 360°).
• Sector: the region bounded by two radii and the arc between them. A quarter circle is a 90° sector.
• Segment: the region bounded by a chord and its minor arc — the smaller leftover piece when a sector loses its triangular center.
• Symbols and units: radius r, diameter d = 2r (length units); arc length in length units; sector and segment areas in square units.

Formulas

• Arc length: l = (θ / 360°) × 2πr, where θ is the central angle in degrees.
• Area of a sector: A = (θ / 360°) × πr² — the same fraction θ/360 of the full circle's area.
• Area of a segment: sector area − triangular area = (θ / 360°) × πr² − (1/2)r²sin θ, for the minor segment.
• Chord length: for a chord subtending central angle θ, chord = 2r sin(θ/2).

Worked Example

A circle has radius 10 cm. Find the arc length and sector area for a central angle of 60°. Use π = 3.14.

1. Arc length: l = (60 / 360) × 2 × 3.14 × 10 = (1/6) × 62.8 = 10.47 cm (rounded to two decimals).
2. Sector area: A = (60 / 360) × 3.14 × 10² = (1/6) × 314 = 52.33 cm².
3. Sanity check: 60° is one-sixth of a full circle, so both answers should be one-sixth of the circumference (62.8 cm) and the full area (314 cm²). They are. Correct.

Common Misconceptions

• A chord is straight; an arc is curved. The chord is the straight cut across; the arc is the bend of the circle itself.
• Sector and segment differ. The sector includes the center like a pizza slice; the segment is the smaller cap cut off by the chord, without the center.
• The fraction must use the central angle. Using any other angle in θ/360 gives a wrong arc length — the vertex must be at the circle's center.

Real-World Applications

• Food: pizza slices and cake portions are sectors, and their sizes are compared using sector area.
• Engineering: gears, pulleys, and wheels use arc length to convert rotation into distance travelled.
• Data visualization: pie charts are literally divided into sectors whose central angles are proportional to the data values.
• Agriculture: center-pivot irrigation waters a circular (or sector-shaped) field, and farmers compute the watered area with the sector formula.

Chords are straight cuts across a circle, arcs are curved stretches of its circumference, and sectors are the pie slices between two radii. Arc length and sector area are both found by taking the fraction θ/360 of the full circumference or area. Keep straight and curved parts distinct, and circle measurement becomes simple.