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Grade 10 · Core Mathematics · Lesson 7 of 12

Area of a Segment of a Circle in Context

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners calculate the area of a minor segment in real-world contexts using sector and triangle formulas.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L740 minutes · 3 slides

KICD learning outcome (d)

Work out the area of a segment of a circle.

By the end of this lesson, learners should be able to

  1. Calculate the area of a minor segment in real-world contexts using sector and triangle formulas.
  2. Determine the area of a major segment by subtracting the minor segment from the total circle area.
  3. Solve practical segment area problems involving given chord lengths and radii.

Key vocabulary

Minor segment
The smaller region of a circle bounded by a chord and its corresponding arc.
The area of the minor segment bounded by the central angle of 72° is 16.87 m².
Major segment
The larger region of a circle bounded by a chord and its corresponding arc.
Subtracting the minor segment area from total circle area gives the major segment area of 686.55 cm².

Lesson preview

Slide 1 · Area of Minor Segments in Architectural ContextsOpening · 12 min
A sector of a circle with a radius of 10.5 metres and a central angle of 72 degrees showing a shaded minor segment.
Figure 1: A sector of a circle with a radius of 10.5 metres and a central angle of 72 degrees showing a shaded minor segment.

In architectural designs and surveillance applications, circular boundaries often create minor segments. To calculate the area of a minor segment, we first find the area of the sector enclosed by the central angle and subtract the area of the triangle formed by the two radii and the chord. Precision in applying trigonometry ensures exact floor area measurements.

  • A minor segment area equals sector area minus the central triangle area.
  • Calculate sector area using the central angle fraction multiplied by πr².
  • Calculate triangle area using the formula ½ r² sin θ.
  • Subtract triangle area from sector area to determine the segment region.
Worked example. Sector area = (72/360) × (22/7) × (10.5)² = 0.2 × 346.5 = 69.3 m². Triangle area = 0.5 × 10.5 × 10.5 × sin 72° = 0.5 × 110.25 × 0.9511 = 52.429 m². Segment area = Sector area − Triangle area = 69.3 − 52.429 = 16.871 m² ≈ 16.87 m². Answer: 16.87 m².

Check question: A police helicopter surveys a sector-shaped region of radius 14 km with a central angle of 60°. Calculate the area of the minor segment bounded by the chord connecting the arc ends. (Take π = 3.142 and sin 60° = 0.8660)

  1. 17.77 km²
  2. 84.87 km²
  3. 102.64 km²
  4. 187.51 km²
Show answer

17.77 km². Sector area = (60/360) × 3.142 × 14² = 102.639 km². Triangle area = 0.5 × 14² × 0.8660 = 84.868 km². Minor segment area = Sector area − Triangle area = 102.639 − 84.868 = 17.771 km² ≈ 17.77 km².

Slide 2 · Finding Major Segment Area from Circle and Minor SegmentModel · 14 min
A circle divided by a chord into a smaller grey minor segment and a larger blue major segment.
Figure 2: A circle divided by a chord into a smaller grey minor segment and a larger blue major segment.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • All 3 slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 11 lessons in this sub-strand
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All 12 lessons in Area of a Part of a Circle

  1. 1Area of an AnnulusDetermine the area of an annulus in different situations
  2. 2Area of an Annulus in Applied ContextsDetermine the area of an annulus in different situations
  3. 3Area of a Sector of a Circle in Real-Life SituationsWork out the area of a sector of a circle in real-life situations
  4. 4Area of a Sector of a Circle in Real-Life SituationsWork out the area of a sector of a circle in real-life situations
  5. 5Area of an Annular SectorDetermine the area of an annular sector in different situations
  6. 6Area of a Segment of a CircleWork out the area of a segment of a circle
  7. 7Area of a Segment of a Circle in ContextWork out the area of a segment of a circle
  8. 8Area of Common Region Between Two Intersecting CirclesDetermine the area of common region between two intersecting circles
  9. 9Applications of Area of Parts of a CircleApply the area of a part of a circle in real-life situations
  10. 10Applying Area of Parts of a Circle in Real-Life SituationsApply the area of a part of a circle in real-life situations
  11. 11Exploring the Area of a Sector in Real-Life ContextsExplore the use of the area of a part of a circle in real-life situations
  12. 12Applications of Area of a Part of a CircleExplore the use of the area of a part of a circle in real-life situations