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Learning Resources

Area of a Part of a Circle

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 10 Pathway: N/A

First Principles

Learning Objective: Understand how sectors and segments are formed and master the calculation of their areas using proportional reasoning and trigonometry.

Total Circle Area: 314.16 cm²

Fraction (θ/360): 0.2500

Sector Area (Blue): 78.54 cm²

Triangle Area (Dotted): 50.00 cm²

Segment Area (Green): 28.54 cm²

1. Understanding the Sector: Think of a circular chapati or pancake of radius \(r\). The full circle has an area of \(\pi r^2\) and spans \(360^\circ\). When you cut out a slice with a central angle \(\theta\), it represents the fraction \(\frac{\theta}{360}\) of the complete circle.

2. Understanding the Segment: A segment is the region bounded by a chord and the corresponding arc. Notice that the sector is made of two components: the inner isosceles triangle formed by the two radii and the chord, plus the curved segment cap.

\[ \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \]

Key Formulas

Total Area of a Circle: \[ A_{\text{circle}} = \pi r^2 \] Where \(r\) is the radius of the circle.
Area of a Sector (Angle in Degrees): \[ A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \] Where \(\theta\) is the central angle in degrees.
Area of a Sector (Angle in Radians): \[ A_{\text{sector}} = \frac{1}{2} r^2 \theta \] Where \(\theta\) is the central angle in radians.
Area of the Triangle inside the Sector: \[ A_{\text{triangle}} = \frac{1}{2} r^2 \sin \theta \] Using the SAS area formula \(\frac{1}{2} a b \sin C\) with \(a = b = r\).
Area of a Minor Segment: \[ A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{\theta}{360^\circ}\pi r^2 - \frac{1}{2} r^2 \sin \theta \]

Worked Examples

Example 1 (Easy): Sector Area of a Circular Farm Plot

A farmer in Machakos has a circular plot irrigated by a center-pivot system of radius \(r = 14\text{ m}\). One section planted with sukuma wiki subtends an angle of \(45^\circ\) at the center. Find the area of this sector. (Take \(\pi = \frac{22}{7}\))

  1. Identify given values: \(r = 14\text{ m}\), \(\theta = 45^\circ\).

  2. Set up the formula: \[ A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \]

  3. Substitute and simplify: \[ A_{\text{sector}} = \frac{45}{360} \times \frac{22}{7} \times 14^2 = \frac{1}{8} \times \frac{22}{7} \times 196 \] \[ A_{\text{sector}} = \frac{1}{8} \times 22 \times 28 = \frac{616}{8} = 77\text{ m}^2 \]

  4. Final Answer: \(\boxed{77\text{ m}^2}\)

Example 2 (Medium): Area of a Segment with a Right Central Angle

A circular tabletop of radius \(10\text{ cm}\) has a decorative glass piece fitting into a minor segment whose central angle is \(90^\circ\). Calculate the area of this segment. (Use \(\pi = 3.14\))

  1. Find the Sector Area: \[ A_{\text{sector}} = \frac{90}{360} \times \pi \times 10^2 = \frac{1}{4} \times 3.14 \times 100 = 78.5\text{ cm}^2 \]

  2. Find the Triangle Area: \[ A_{\text{triangle}} = \frac{1}{2} r^2 \sin(90^\circ) = \frac{1}{2} \times 10^2 \times 1 = 50\text{ cm}^2 \]

  3. Subtract Triangle from Sector: \[ A_{\text{segment}} = 78.5 - 50 = 28.5\text{ cm}^2 \]

  4. Final Answer: \(\boxed{28.5\text{ cm}^2}\)

Example 3 (Hard): Segment Area with \(\theta = 60^\circ\)

A circle of radius \(6\text{ cm}\) has a chord subtending an angle of \(60^\circ\) at the center. Find the exact area of the minor segment, and give its value to 2 decimal places. (Use \(\pi = 3.142\), \(\sqrt{3} = 1.732\))

  1. Calculate Sector Area: \[ A_{\text{sector}} = \frac{60}{360} \times \pi \times 6^2 = \frac{1}{6} \times 3.142 \times 36 = 6 \times 3.142 = 18.852\text{ cm}^2 \]

  2. Calculate Triangle Area: \[ A_{\text{triangle}} = \frac{1}{2} r^2 \sin(60^\circ) = \frac{1}{2} \times 36 \times \frac{\sqrt{3}}{2} = 9\sqrt{3} \] \[ A_{\text{triangle}} = 9 \times 1.732 = 15.588\text{ cm}^2 \]

  3. Calculate Minor Segment Area: \[ A_{\text{segment}} = 18.852 - 15.588 = 3.264\text{ cm}^2 \approx 3.26\text{ cm}^2 \]

  4. Final Answer: \(\boxed{3.26\text{ cm}^2}\)

Common Mistakes

Mistake 1: Confusing Arc Length with Sector Area

Why students do it Both formulas use the fraction \(\frac{\theta}{360^\circ}\). Students mistakenly multiply by circumference (\(2\pi r\)) when asked for area.

Correction Always check units! Arc length measures distance (cm, m) using \(\frac{\theta}{360^\circ} \times 2\pi r\), while sector area measures 2D surface (cm², m²) using \(\frac{\theta}{360^\circ} \times \pi r^2\).

Mistake 2: Forgetting to Convert Diameter to Radius

Why students do it Questions frequently provide the diameter (e.g., "a wheel of diameter 14 cm"), and students plug \(d\) directly into \(\pi r^2\).

Correction Always write \(r = \frac{d}{2}\) as your very first step before squaring.

Mistake 3: Confusing Sectors and Segments

Why students do it The words sound similar. A sector is like a slice of cake/pizza (bounded by 2 radii and an arc). A segment is bounded by a straight chord and an arc.

Correction Remember: Segment = Sector − Triangle.

Real World

Center-Pivot Irrigation in Kenya: In large-scale agricultural projects (like in Naivasha or Kibwezi), rotating sprinklers water sectors of farmland. Farmers calculate the sector area to calibrate seed density and fertilizer quantity.
Traditional Kenyan Architecture: Circular huts and thatched roofs often feature triangular trusses supporting circular segment eaves. Calculating the area helps determine timber and thatch requirements.
Road and Highway Engineering: Curved bends and roundabout exits are engineered using circular arcs and segments to determine asphalt paving areas and drainage slopes.
Solar and Wind Technology: A wind turbine blade sweeps out a circular area; when yaw motors restrict rotation to a sector arc during high-wind safety protocols, engineers calculate the swept sector area to predict power output.

Practice

A sector of a circle has a radius of 7 cm and a central angle of 90 degrees. Find the area of the sector in square centimetres. (Use pi = 22/7) (Type only the number, e.g., 42.5)
Review the concepts above.
Leshore, a carpenter in Nakuru, cuts a sector-shaped piece of wood for a table top. The sector has a radius of 10 cm and a central angle of 72 degrees. What is its area in square centimetres? (Use pi = 3.14) (Type only the number, e.g., 45.6)
Review the concepts above.
A community flower garden in Nairobi is circular with a diameter of 14 metres. The school wants to pave a sector of the garden that subtends an angle of 60 degrees at the centre. Find the area of this sector in square metres to two decimal places. (Use pi = 22/7) (Type only the number, e.g., 25.67)
Review the concepts above.
A sector of a circle of radius 6 cm has a central angle of 90 degrees. Find the area of the minor segment (in square centimetres) bounded by the chord connecting the endpoints of the radii. (Use pi = 3.14) (Type only the number, e.g., 10.26)
Review the concepts above.
A circular plot of radius 12 m has a fence chord subtending an angle of 60 degrees at the center. Find the area of the minor segment in square metres. (Use pi = 3.142 and sqrt(3) = 1.732. Round your answer to 2 decimal places.) (Type only the number, e.g., 13.06)
Review the concepts above.
A stained-glass window design includes a circular segment in a circle of radius 10 cm. The chord subtends a central angle of 120 degrees. Calculate the area of the minor segment in square centimetres. (Use pi = 3.142 and sin(120 degrees) = 0.866. Round your answer to 2 decimal places.) (Type only the number, e.g., 61.43)
Review the concepts above.