Learning Objectives
  • Calculate minor and major arc lengths.
  • Calculate minor and major sector areas.
  • Work backwards to find a central angle or radius.
  • Combine sectors, triangles and straight lengths in perimeter and area problems.
  • Give exact answers in terms of \pi or suitable decimal approximations.
Key Terms
Arc
A portion of the circumference of a circle.
Sector
A region bounded by two radii and an arc.
Central angle
The angle at the centre subtended by an arc.
Minor arc
The shorter arc between two points.
Major arc
The longer arc between two points.
What This Chapter Covers
  • Arc length as a fraction of circumference
  • Sector area as a fraction of circle area
  • Major and minor sectors
  • Compound perimeters and areas
  • Inverse arc and sector calculations
Fractions Of A Circle

A central angle of \theta^\circ represents the fraction \theta/360 of a full circle. Multiply this fraction by circumference for arc length or by circle area for sector area.

\text{arc length}=\frac{\theta}{360^\circ}\times2\pi r
\text{sector area}=\frac{\theta}{360^\circ}\times\pi r^2
Worked Example: Minor Arc And Sector

Question: A sector has radius 9 cm and angle 80^\circ. Find its arc length and area.

  1. Arc =\frac{80}{360}\times2\pi(9)=4\pi.
  2. Area =\frac{80}{360}\times\pi(9^2)=18\pi.

Answer: Arc 4\pi cm; area 18\pi cm².

Major Arcs And Major Sectors

The major angle is 360^\circ-\theta. Alternatively subtract the minor arc or sector from the full circumference or circle area.

Worked Example: Major Sector

Question: A minor sector has angle 110^\circ in a circle of radius 6 cm. Find the major-sector area.

  1. Major angle =360-110=250^\circ.
  2. Area =\frac{250}{360}\pi(6^2).

Answer: 25\pi cm².

Perimeter Of A Sector

The perimeter of a sector includes the arc and two radii. Do not give arc length alone.

\text{sector perimeter}=\text{arc length}+2r
Worked Example: Sector Perimeter

Question: Find the perimeter of a 120^\circ sector of radius 7 cm.

  1. Arc =\frac{120}{360}\times14\pi=\frac{14\pi}{3}.
  2. Add two radii: 14.

Answer: 14+\frac{14\pi}{3}\approx28.7 cm.

Working Backwards
Worked Example: Find The Angle

Question: An arc has length 5\pi cm in a circle of radius 12 cm. Find the central angle.

  1. Use 5\pi=\frac{\theta}{360}\times24\pi.
  2. Cancel \pi and solve: 5=\theta/15.

Answer: \theta=75^\circ.

Examination Guidance
  • Decide whether the question requires a minor or major arc.
  • For sector perimeter, add both radii.
  • Keep \pi exact when requested.
  • Use the central angle, not an angle at the circumference.
Common Mistakes
  • Using 180 instead of 360 in degree formulas.
  • Confusing sector area with arc length.
  • Forgetting the two radii in a sector perimeter.
  • Using the minor angle for a major sector.
Knowledge Check And Practice

1. Find the arc length of a 90^\circ sector of radius 8 cm.

Answer: 4\pi cm.

2. Find the area of a 45^\circ sector of radius 12 cm.

Answer: 18\pi cm².

3. Find the major angle corresponding to a minor angle of 135^\circ.

Answer: 225^\circ.

4. A semicircle has radius 5 cm. Find its curved arc length.

Answer: 5\pi cm.

5. What extra lengths are included in a sector perimeter?

Answer: Two radii.
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