Grade 10 essential mathematics Measurements and Geometry – Area of a Part of a Circle Notes

1. (a) Identify and outline the sub-sub-strands: Area of a sector and Area of a segment.
2. (b) Determine the area of a sector of a circle in different situations (degrees and radians).
3. (c) Determine the area of a segment of a circle in different situations (minor and major segments).
4. (d) Apply the area of part of a circle in real-life situations (e.g., pizza slice, roundabout, pond section, tank covers).
5. (e) Appreciate the application of the area of parts of circular objects in real life.

• Circle area: A = πr² (r = radius).
• Area of a sector with central angle θ:
  • If θ is in degrees: Area = (θ/360) × πr².
  • If θ is in radians: Area = (1/2) r² θ.
• Area of a segment (region between a chord and arc):
  • Find area of sector with the same central angle, then subtract area of the triangle formed by the two radii and the chord.
  • If θ in degrees:
    Area of triangle = (1/2) r² sin(θ°) (use sin function in degrees).
    Segment area = (θ/360)πr² − (1/2) r² sin(θ°).
  • If θ in radians:
    Area of triangle = (1/2) r² sin θ (θ in radians).
    Segment area = (1/2) r² θ − (1/2) r² sin θ = (1/2) r² (θ − sin θ).
• Minor vs major segment: If θ > 180° the sector gives the major sector. For the minor segment use the smaller central angle (360° − θ) or compute major and subtract from full circle area.

• Designing a circular flower bed in school grounds: compute area of a sector if only part is planted.
• Road works — painting and refurbishing roundabout sectors or island segments.
• Covering a circular water tank where part of the lid is removed — compute material needed for a sector or segment.
• Food industry — slices of chapati, pizza, or cake — determining serving area per slice.
• Agriculture — irrigation of a circular field where a straight fence (chord) cuts off a section; find area of the remaining segment.

Knowing how to work with parts of circles helps in estimating materials, costs and quantities when dealing with rounded objects and spaces common in everyday Kenyan life (tanks, roundabouts, dishes, garden beds).

• Start with concrete objects: bring round plates, lids, pans. Use a string to find diameter and measure radius, then mark sectors with a protractor and cut paper sectors.
• Group task: each group measures a circular object (e.g., saucepan lid), chooses an angle (30°, 45°, 90°), and calculates sector area; cut a paper model to verify.
• Field task: visit a nearby roundabout or circular water-tank. Measure radius visually/approx, estimate sectors/segments formed by a road or fence and compute areas. Discuss accuracy and sources of error.
• Problem solving: given sector area & radius, find central angle; or given segment area & radius, set up equation and approximate θ numerically (use calculator for sin or radians).
• Project: design a circular garden bed with a paved chord across it — compute area of planted (segment) and paved area. Present materials needed and cost estimate (local context: seeds, topsoil, paving stones).
• Use calculators in degree and radian modes; practise converting degrees ↔ radians: radians = degrees·π/180.

1. r = 6 cm, θ = 45°. Find the area of the sector. (Answer: (45/360)π·36 = (1/8)·36π = 4.5π ≈ 14.14 cm².)
2. r = 10 cm, θ = 150°. Find the area of the segment. (Sector = (150/360)π·100 = (5/12)·100π = 125π/3 ≈130.9. Triangle = 1/2·100·sin150° = 50·0.5 =25. Segment ≈130.9−25 =105.9 cm².)
3. A pizza has radius 12 cm. One slice (sector) has central angle 30°. What fraction of the pizza is the slice and what is its area? (Fraction = 30/360 = 1/12. Area = (1/12)π·144 =12π ≈37.70 cm².)
4. r = 4 m, θ = π/2 rad. Find sector area using radians. (Area = 1/2·16·π/2 = 8·π/2 = 4π ≈12.57 m².)

• Short test: calculate sector areas and segment areas with angles in degrees and radians.
• Practical test: measure a circular object and estimate a sector area, compare with calculation.
• Project assessment: group presentation of a design using sector/segment areas (garden bed, shade canopy, roundabout painting) including calculations and a simple cost estimate.

• Sector area formula works in degrees or radians — be careful with units.
• Segment area = sector area − triangle area (the triangle formed by two radii and chord).
• Use the smaller central angle to get the minor segment (or compute major by subtraction from circle area).
• Applications include everyday Kenyan contexts: cooking, town planning, farming and water storage.