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Lesson 9 of 10

Measuring Space: Perimeter and Area · Lesson 9 of 10

Area of Sectors and Segments

“Sectors take the slice while segments keep the curved leftovers.”

Learning Objectives

• Define a sector and a segment. • Derive area of a sector from central angle. • Calculate semicircle and quadrant areas. • Distinguish minor and major sectors and segments. • Find segment area by subtracting triangle area from sector area.

Sector

A sector is the part of a circle enclosed by two radii and the arc between their endpoints. You can think of it as a slice of the circle. The two radii meet at the centre and form an angle, while the curved boundary of the sector is the corresponding arc. The size of the sector depends on the angle formed by the two radii.

Definition
Sector

The region bounded by two radii and the arc joining their endpoints.

A sector occupies the same fraction of a circle's area as its central angle occupies of a complete 360° turn.

Sector areaLaTeX
Sector with central angle theta A circle with centre O, radii OA and OB, a shaded sector AOB, and central angle AOB labelled theta. Sector AOB θ A B O OA OB
Sector with central angle theta

Special Cases

Semicircle areaLaTeX
Quadrant areaLaTeX
Worked Example: 60° Sector

Problem
Find the area of a 60° sector in a circle of radius 7 cm using π≈22/7.

  1. 1.A=πr²×60/360.
  2. 2.A=22/7×49×1/6.
  3. 3.A=77/3 cm².

Minor and Major Sectors

A chord and its endpoints determine a smaller sector and a larger sector. If the minor central angle is θ, the major central angle is 360°−θ.

Segments

Definition
Segment of a Circle

The region bounded by a chord and the corresponding arc.

Minor Sector and Minor Segment Shaded regions Minor sector Region bounded by the two radii and the minor arc. Minor segment Region between the chord and the minor arc. The chord and radii are shown only to identify the regions.
Minor sector and minor segment

For a minor segment, we can often find its area by taking the sector area and subtracting the area of triangle AOB.

Minor segment areaLaTeX
Worked Example: Clock Sweep

Problem
A minute hand is 7 cm long. Find the area swept in 10 minutes.

  1. 1.In 60 minutes the hand turns 360°.
  2. 2.In 10 minutes it turns 60°.
  3. 3.Swept area=π×7²×60/360.
  4. 4.Using 22/7, area=77/3 cm².
Worked Example: Wiper Area

Problem
A wiper of length 28 cm sweeps through 120°. Find the area cleaned by one sweep.

  1. 1.Treat the swept region as a sector of radius 28 cm.
  2. 2.A=π×28²×120/360.
  3. 3.A=(1/3)π×784.
  4. 4.Using 22/7, A=2464/3 cm².

Practice Problems

Practice Problems
  1. Find the area of a quadrant whose circle has circumference 44 cm.
  2. A chord of a circle of radius 10 cm subtends 90° at the centre. Find the minor and major sector areas.
  3. A chord in a circle of radius 15 cm subtends 60°. Find the minor sector area.
  4. A 60° chord in a radius-r circle forms an equilateral triangle with the centre. Express the minor segment area in terms of r.
  5. Two non-overlapping wipers each have length 28 cm and sweep 120°. Find the total area cleaned.

Key Takeaways

Key Takeaways

• Sector area is θ/360 of πr². • A semicircle is half the circle; a quadrant is one quarter. • A segment is bounded by a chord and an arc. • Minor segment area is often sector area minus triangle area. • Clock hands and wipers are natural sector-area applications.

Coming Next

Next, we revise the complete chapter and practise choosing the correct perimeter or area method.

Previous · Lesson 8

Area of a Circle

Next · Lesson 10

Chapter Summary and Practice