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Learning Resources

Area of 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

Objective: Understand and calculate the area of a sector of a circle from first principles using proportional reasoning in both degrees and radians.

Imagine a circular communal shamba (farm) in Naivasha or a large round Chapati. When sliced from the centre to the edge along two radii, each piece formed is called a sector. Since the full circle represents a complete turn of \(360^\circ\) (or \(2\pi\) radians), a sector with central angle \(\theta\) is simply a fraction of the entire circle's area.

(a) Fractional Area Proportions

A circle of radius \(r\) has a total area given by \(A_{\text{total}} = \pi r^2\). If you cut out an angle \(\theta^\circ\), the fraction of the total turn is:

\[ \text{Fraction of Circle} = \frac{\theta}{360^\circ} \]

Therefore, the area of the sector is directly proportional to its central angle:

\[ A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \]

(b) Interactive Sector Explorer

Adjust the radius and central angle below to see how the sector wedge scales relative to the whole circle.

Interactive Sector Area Visualizer

Key Formulas

1. Area of a Sector (Degrees)

\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]

where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.

2. Area of a Sector (Radians)

\[ A = \frac{1}{2} r^2 \theta \]

where \(\theta\) is the central angle in radians. Derived from \(\frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2} r^2 \theta\).

3. Relation Between Arc Length and Sector Area

\[ A = \frac{1}{2} L r \]

where \(L = \frac{\theta}{360^\circ}(2\pi r) = r\theta\) is the arc length enclosing the curved boundary of the sector.

4. Perimeter of a Closed Sector

\[ P = L + 2r = \left(\frac{\theta}{360^\circ} \times 2\pi r\right) + 2r \]

A closed sector boundary consists of the curved arc plus the two straight straight bounding radii.

Worked Examples

Example 1 (Easy): Basic Sector Calculation

Problem: A circular irrigation sprinkler rotates through an angle of \(90^\circ\) with a spray range (radius) of \(14\text{ m}\). Using \(\pi = \frac{22}{7}\), find the area watered by the sprinkler.

  1. Identify formula: \(A = \frac{\theta}{360^\circ} \times \pi r^2\).
  2. Substitute given values: \(\theta = 90^\circ\), \(r = 14\text{ m}\), \(\pi = \frac{22}{7}\). \[ A = \frac{90^\circ}{360^\circ} \times \frac{22}{7} \times (14)^2 \]
  3. Simplify: \(\frac{90}{360} = \frac{1}{4}\) and \(14^2 = 196\). \[ A = \frac{1}{4} \times \frac{22}{7} \times 196 = \frac{1}{4} \times 22 \times 28 = \frac{616}{4} = 154\text{ m}^2 \]
  4. Answer: The watered sector area is \(154\text{ m}^2\).

Example 2 (Medium): Sector Area from Arc Length

Problem: A decorative Maasai beaded disc sector has a radius of \(10\text{ cm}\) and an outer arc length of \(15\text{ cm}\). Calculate the area of this sector.

  1. Select direct relation: When arc length \(L\) and radius \(r\) are known, use \(A = \frac{1}{2} L r\).
  2. Substitute values: \(L = 15\text{ cm}\), \(r = 10\text{ cm}\). \[ A = \frac{1}{2} \times 15 \times 10 \]
  3. Compute: \[ A = 15 \times 5 = 75\text{ cm}^2 \]
  4. Answer: The area of the beaded sector is \(75\text{ cm}^2\).

Example 3 (Hard): Finding Central Angle from Area and Reverse Problem

Problem: A solar panel installer in Machakos cuts a sector of radius \(21\text{ cm}\) from a circular solar sheet. The area of the sector is \(462\text{ cm}^2\). Taking \(\pi = \frac{22}{7}\), find:
(a) The central angle \(\theta\) in degrees.
(b) The total perimeter of this cutout sector.

  1. Step 1: Determine the total circle area: \[ A_{\text{total}} = \pi r^2 = \frac{22}{7} \times 21^2 = \frac{22}{7} \times 441 = 22 \times 63 = 1386\text{ cm}^2 \]
  2. Step 2: Solve for central angle \(\theta\): \[ \frac{\theta}{360^\circ} \times 1386 = 462 \implies \frac{\theta}{360^\circ} = \frac{462}{1386} = \frac{1}{3} \] \[ \theta = \frac{1}{3} \times 360^\circ = 120^\circ \]
  3. Step 3: Calculate the perimeter of the sector: \[ L = \frac{\theta}{360^\circ} \times 2\pi r = \frac{1}{3} \times 2 \times \frac{22}{7} \times 21 = \frac{1}{3} \times 132 = 44\text{ cm} \] \[ P = L + 2r = 44 + 2(21) = 44 + 42 = 86\text{ cm} \]
  4. Answer: Central angle is \(120^\circ\) and total perimeter is \(86\text{ cm}\).

Common Mistakes

1. Mistake: Confusing Diameter with Radius in Area Calculations

Why it feels right The diameter is often given on diagrams across the whole circular base (e.g., a \(14\text{ m}\) diameter tank).

Correction Always halve the diameter first to find radius: \(r = \frac{d}{2}\). Using diameter directly in \(\pi d^2\) overestimates the area by a factor of 4!

2. Mistake: Forgetting the Straight Boundaries in Sector Perimeter

Why it feels right Students often calculate only the arc length \(L = \frac{\theta}{360}(2\pi r)\) and assume that is the perimeter of the sector.

Correction A sector is enclosed by the curved arc AND two straight radii. Therefore, \(\text{Perimeter} = L + 2r\).

3. Mistake: Mixing Degrees and Radians in Formulae

Why it feels right Students see \(A = \frac{1}{2}r^2\theta\) and plug in \(\theta = 60\), getting a huge incorrect area.

Correction The formula \(\frac{1}{2}r^2\theta\) applies ONLY when \(\theta\) is in radians. If \(\theta\) is in degrees, either use \(\frac{\theta}{360}\pi r^2\) or convert degrees to radians first using \(\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\).

Real World

Centre-Pivot Irrigation in Kibwezi: Rotating sprinkler arms sweep out sectors across large farmlands. Agribusiness managers calculate sector areas to determine the exact volume of fertilizer (fertigation) needed for specific crop zones.
Road Roundabouts & Civil Engineering: Traffic engineers in Nairobi design slip roads and curved lane dividers by calculating sector and arc dimensions to manage turning radiuses for heavy matatus and trucks.
Renewable Energy (Wind Turbines): As wind turbine blades rotate through a partial arc during directional adjustment, aerodynamic engineers compute the swept sector area to calculate the instantaneous kinetic wind power intercepted.
Fabric and Tailoring (Kitenge & Flare Skirts): Circular flare skirts (umbrella cuts) are made by cutting annular sectors out of kitenge fabric. Calculating the sector area ensures minimal fabric wastage during cutting.

Practice

A matatu company logo is a circle with a central sector painted yellow. The sector has a central angle of 135°. What fraction of the full circle's area is painted yellow in simplest fractional form? (Type only the fraction, e.g., 3/8)
Review the concepts above.
Mukabi, a tailor, cuts a sector-shaped cloth piece with radius 21 cm and a central angle of 60°. Using \(\pi = \frac{22}{7}\), calculate the area of the cloth sector in cm². (Type only the number, e.g., 231)
Review the concepts above.
A circular lawn sprinkler has a reach radius of 12 m and rotates through a central angle of 120°. Calculate the arc length along the outer boundary swept by the water spray in metres. Use \(\pi = 3.142\) and round to one decimal place. (Type only the number, e.g., 25.1)
Review the concepts above.
A sector of a circle has a radius of 10 cm and an arc length of 14 cm. Calculate the area of the sector in cm². (Type only the number, e.g., 70)
Review the concepts above.
A cattle feeding trough is in the shape of a sector of radius 14 m. The area of the sector is 77 m². Taking \(\pi = \frac{22}{7}\), calculate the central angle \(\theta\) of the trough in degrees. (Type only the number, e.g., 45)
Review the concepts above.
A fenced sector-shaped vegetable garden has a central angle of 90° and an area of 154 m². Taking \(\pi = \frac{22}{7}\), find the total length of fencing required to enclose the entire perimeter of the sector in metres. (Type only the number, e.g., 50)
Review the concepts above.