Grade 10 core mathematics Measurements and Geometry – Area of a Part of a Circle (12 lessons) Notes

This unit (12 lessons) develops students' ability to compute and apply areas of parts of circles: annulus (ring), sector, annular sector, segment, and the common region (lens) of two intersecting circles. Emphasis is on understanding, deriving formulas, solving real-life problems and using metric units appropriate to Kenya (mm, cm, m).

1. Identify and outline the sub-sub-strands:
   • Area of an annulus
   • Area of a sector of a circle
   • Area of an annular sector
   • Area of a segment of a circle
   • Area of the common region (lens) between two intersecting circles
   • Applications of areas of parts of a circle
2. Determine the area of an annulus in different situations.
3. Compute the area of a sector in real-life situations.
4. Find the area of an annular sector in different situations.
5. Work out the area of a segment of a circle (using sector minus triangle).
6. Determine the area of the common region between two intersecting circles (lens).
7. Apply these areas to practical Kenyan contexts (e.g., ring-shaped garden beds, circular windows, pizza slices, manhole covers).
8. Explore uses of these areas in projects and investigations.

• Area of a circle: A = π r²
• Area of annulus (ring) with outer radius R and inner radius r: A = π (R² − r²)
• Sector area with central angle θ (degrees): A = (θ/360) · π r². In radians: A = ½ r² θ.
• Annular sector (between two concentric radii R and r, angle θ): A = (θ/360) · π (R² − r²)
• Segment (central angle θ in degrees): Area(segment) = Area(sector) − Area(triangle)
  For an isosceles triangle with sides r,r and included angle θ: triangle area = ½ r² sin θ (θ in radians or convert sin for degrees).
• Common area (lens) of two circles radius r and R with centers distance d:
  A = r² cos⁻¹((d² + r² − R²)/(2 d r)) + R² cos⁻¹((d² + R² − r²)/(2 d R)) − ½ √{(−d+r+R)(d+r−R)(d−r+R)(d+r+R)}
  (Use calculators; derive from sum of two circular segments.)
• Use π ≈ 3.142 or 22/7 where appropriate; indicate which approximation is used.

1. Lesson 1 — Review of circle basics (1 lesson)
   • Objectives: recall radius, diameter, circumference, area of circle, units.
   • Activity: measure lids and compute areas; check units (cm²).
2. Lesson 2 — Annulus (1 lesson)
   • Outline annulus and derive A = π(R² − r²).
   • Classwork: ring-shaped flowerbed examples (outer R, inner r).
3. Lesson 3 — Sector — definition and formula (1 lesson)
   • Derive sector area from fraction of circle: A = (θ/360)πr².
   • Group activity: slices of pizza, water tank sector.
4. Lesson 4 — Annular sector (1 lesson)
   • Combine annulus and sector ideas: A = (θ/360)π(R² − r²).
   • Practical: ring-shaped road markings or circular garden beds.
5. Lesson 5 — Segment — concept and formula (1 lesson)
   • Segment = sector − triangle. Show triangle area = ½ r² sin θ (θ in radians) or derive using ½ab sin C.
   • Example: area of a shallow pond cap cut by a straight path.
6. Lesson 6 — Segment problems (1 lesson)
   • Practice: given r and θ (degrees), compute segment area. Emphasize calculator use for sin and cos and unit practice.
7. Lesson 7 — Intersecting circles: sketch and reasoning (1 lesson)
   • Show two circles overlap; area of overlap = sum of two circular segments.
   • Derive expressions using cos⁻¹ and triangle area (general formula provided).
8. Lesson 8 — Intersecting circles: worked examples (1 lesson)
   • Examples with equal radii (simpler) and unequal radii; use calculators; check special cases (no overlap, one inside another).
9. Lesson 9 — Applications: real life contexts (1 lesson)
   • Problems: cover for a circular tank with co-axial hole (annulus), pizza slices, circular beams cutouts, traffic roundabout markings.
10. Lesson 10 — Mixed problem solving (1 lesson)
    • Higher-order problems combining several parts (e.g., annular sector cut by chord forming segments).
11. Lesson 11 — Written test / summative assessment (1 lesson)
    • Test covering all sub-strands; include practical word problems and an investigative question.
12. Lesson 12 — Project / consolidation and exploration (1 lesson)
    • Group project: measure real objects (manhole cover, ring-shaped garden), model, compute areas, present findings and usage.

• Hands-on measuring: bring lids, tyres, plates, manhole covers; measure radii and compute areas of circle and annulus.
• Group experiment: mark a circular garden bed with inner hole (annulus) and calculate area of soil to be planted — connect to agriculture class.
• Real-life problem: design a circular window with a decorative ring (annular sector) — compute glass area and frame area for budgeting.
• Use simple geometry software (GeoGebra) or a scientific calculator for segments and lens areas; show step-by-step calculator key presses for inverse cosine and square roots.
• Set short projects: measure local infrastructure items (manhole cover, water tank) and prepare a one-page report including diagrams and area calculations.
• Encourage use of units: convert mm² ↔ cm² ↔ m² where necessary; discuss appropriate unit for area in each problem.

1. Find the area of an annulus with outer diameter 30 cm and inner diameter 18 cm. (Give exact and approximate answers.)
2. A circular pizza radius 12 cm is cut into a sector of angle 120°. Find the area of this slice.
3. An annular sector has inner radius 3 m, outer radius 6 m and central angle 90°. Find its area.
4. Find the area of the segment of a circle with radius 7 cm cut off by a chord that subtends an angle of 90° at the centre.
5. Two circles of radii 8 cm and 6 cm have centres 10 cm apart. Find the area of their overlap (use calculator and show steps).

Answers (brief): 1) A = π(15²−9²)=π(225−81)=144π cm² ≈ 452.39 cm². 2) (120/360)π(12²)= (1/3)π144 = 48π ≈150.8 cm². 3) (90/360)π(36−9)= (1/4)π27 = 6.75π ≈21.21 m². 4) Sector area=(90/360)π49= (1/4)π49=12.25π≈38.48 cm²; triangle area=½·49·sin90°=24.5; segment ≈13.98 cm². 5) Use formula — calculator required; teacher to provide worked solution in class.

• Formative: quick quizzes on deriving formulas and simple calculations; observe measuring activities.
• Summative (lesson 11): paper with 6–8 questions including at least one practical/contextual problem and one lens/segment problem. Allow calculators for inverse trig and square roots.
• Marking: check units, showing working (derivation of sector/segment), correct use of π and accuracy of approximations.

• Rulers, compasses, protractors, scientific calculators.
• GeoGebra (optional) for dynamic exploration.
• Examples from local context: plates, tyres, water tank lids, circular garden plots.
• Reference: KCSE-style past paper questions on circle parts (teachers to collect relevant past questions).

• Start with visual and physical activities to make abstract areas meaningful.
• Emphasise reasoning: e.g., segment = sector − triangle rather than memorisation only.
• Use simple angles (30°, 60°, 90°, 180°) for initial practice then move to arbitrary angles requiring calculators.
• Relate to other topics (trigonometry, Pythagoras) when finding chord lengths or triangle areas inside circles.