Areas Related to Circles • Topic 2 of 3

Areas of sector of a circle

What is a sector of a circle? A sector is a portion of a circle's interior region bounded by two radii and an arc. Think of a sector as a single slice of a circular pizza cut cleanly from the center point out to the edge.

Sectors come in two distinct sizes:

  • Minor Sector: The smaller slice of the circle, corresponding to an interior angle theta ($\theta$) that is less than 180 degrees.
  • Major Sector: The remaining large piece of the circle left behind, corresponding to an angle equal to $360^\circ - \theta$.

To calculate the properties of a sector, we compare its interior angle $\theta$ to the complete full turn angle of a circle, which is 360 degrees. A sector is simply a fractional piece ($\theta / 360^\circ$) of the entire circle!

Formulas for sector properties:

  • Length of the sector arc (l) = $(\theta / 360^\circ) \cdot 2 \cdot \pi \cdot r$
  • Area of the minor sector = $(\theta / 360^\circ) \cdot \pi \cdot r^2$
  • Area of the major sector = $((360^\circ - \theta) / 360^\circ) \cdot \pi \cdot r^2$

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DIAGRAM 1: PIZZA-SLICE STRUCTURE OF A SECTOR

                    . - - ~ ~ - - .
                .                   .
              /       Major           \
             /        Sector           \
            ;             O             ;
             \           / \           /
              \  Radius /   \ Radius  /
                .      /  θ  \      .
                    . /_ _ _ _\ .
                     Minor Arc
                     [ Minor Sector Slice ]

DIAGRAM 2: MEASURING FRACTIONAL ROTATION

          Quarter Slice (90°)               Half Circle (180°)
               . - ~ ~ - .                       . - ~ ~ - .
             .     |       .                   .           .
            /      |        \                 /             \
           ;_______O         ;               ;_______O_______;
            \               /                 \             /
             .             .                   .           .
               ' - _ _ _ - '                     ' - _ _ _ - '
            Fraction = 90/360                 Fraction = 180/360
                     = 1/4                             = 1/2
Solved Examples
4
Worked Example
Find the area of a sector of a circle with a radius of 6 cm if the angle of the sector is exactly 60 degrees. (Take $\pi = 22/7$).
Solution
  1. Step 1: Identify the given variables.
  2. Radius ($r$) = 6 cm, Angle ($\theta$) = 60 degrees.
  3. Step 2: Use the minor sector area formula.
  4. Area of Sector = $(\theta / 360^\circ) \cdot \pi \cdot r^2$
  5. Area = $(60 / 360) \cdot (22 / 7) \cdot 6 \cdot 6$
  6. Step 3: Simplify the fractional components.
  7. The fraction $60 / 360$ simplifies down to $1 / 6$.
  8. Area = $(1 / 6) \cdot (22 / 7) \cdot 36$
  9. Area = $(1) \cdot (22 / 7) \cdot 6 = 132 / 7$ square cm.
  10. Step 4: Reduce to decimal form if needed.
  11. $132 / 7 = 18.85$ square cm.
  12. Answer: The area of the sector is 132/7 square cm (or approximately 18.85 square cm).
5
Worked Example
An electric windshield wiper on a car has a blade length of 21 cm. As it wipes back and forth, it sweeps through a wide angle of 120 degrees. Find the total area of the glass surface cleaned in a single sweep. (Take $\pi = 22/7$).
Solution
  1. Step 1: Map the wiper blade movement to a circle sector.
  2. The length of the wiper blade acts as the radius: $r = 21$ cm.
  3. The sweeping angle acts as theta: $\theta = 120$ degrees.
  4. Step 2: Set up the calculation using the sector formula.
  5. Cleaned Area = $(\theta / 360^\circ) \cdot \pi \cdot r^2$
  6. Cleaned Area = $(120 / 360) \cdot (22 / 7) \cdot 21 \cdot 21$
  7. Step 3: Simplify the fractions.
  8. The fraction $120 / 360$ simplifies cleanly to $1 / 3$.
  9. Cleaned Area = $(1 / 3) \cdot (22 / 7) \cdot 21 \cdot 21$
  10. Cleaned Area = $(1 / 3) \cdot 22 \cdot 3 \cdot 21$
  11. Step 4: Perform the final multiplication.
  12. The 3 in the denominator cancels with the intermediate 3.
  13. Cleaned Area = $22 \cdot 21 = 462$ square cm.
  14. Answer: The total area cleaned is 462 square cm.
6
Worked Example
A circular wall clock has a minute hand that is 14 cm long. Find the total surface area swept across by the minute hand on the clock face in a time interval of 15 minutes. (Take $\pi = 22/7$).
Solution
  1. Step 1: Calculate the angle turned by the minute hand in 15 minutes.
  2. A full hour consists of 60 minutes, which sweeps a full circle of 360 degrees.
  3. Angle swept per minute = $360^\circ / 60 = 6$ degrees.
  4. Angle swept in 15 minutes ($\theta$) = $15 \cdot 6 = 90$ degrees.
  5. Step 2: Identify the radius.
  6. The length of the minute hand acts as our radius: $r = 14$ cm.
  7. Step 3: Use the sector area equation for a 90-degree angle.
  8. Area Swept = $(90 / 360) \cdot \pi \cdot r^2$
  9. Area Swept = $(1 / 4) \cdot (22 / 7) \cdot 14 \cdot 14$
  10. Area Swept = $(1 / 4) \cdot 22 \cdot 2 \cdot 14$
  11. Area Swept = $(1 / 4) \cdot 616 = 154$ square cm.
  12. Answer: The area swept is 154 square cm.
  13. --

Key Points

  • A sector is a portion of a circle bounded by two radii lines and an outer connecting arc.
  • The area of a sector depends directly on its central interior angle theta ($\theta$).
  • Every sector is calculated as a fraction of a full circle's area: $\frac{\theta}{360^\circ} \cdot \pi r^2$.
  • The length of a sector's curved edge line is called its arc length, calculated as $\frac{\theta}{360^\circ} \cdot 2\pi r$.
  • A minor sector spans an angle less than 180°, while a major sector covers the remaining angle up to 360°.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The area of a sector of angle $\theta$ is:
Explanation: Fraction of the circle's area.
Q2.The length of an arc of angle $\theta$ is:
Explanation: Fraction of the circumference.
Q3.A quadrant of a circle is a sector of angle:
Explanation: A quarter $=90^\circ$.
Q4.A semicircle is a sector of angle:
Explanation: Half the circle.
Practice more MCQs → 📝 Assignment · 35 marks