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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.6.2 Area of a Sector of a Circle

Activity 2.6.3.

\(\textbf{Work in groups}\)
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What you require:
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A graph paper and a razorblade or a pair of scissor✂️
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  1. Draw a circle of radius \(7 \, cm\) on a graph paper.
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  2. Cut out the circle along its boundary.
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  3. Mark the centre of the circle.
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  4. Measure an angle of \(30^\circ\) at the centre and cut out as shown.
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  5. Estimate the area by counting the number of squares enclosed by the arc and the two radii of the circle.
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  6. Express the angle of the sector (\(30^\circ\)) as a fraction of the angle at the centre of the circle (\(360^\circ\)).
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  7. Multiply the fraction obtained in (6) by the area of the circle.
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  8. Discuss and share the result with other groups.
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\(\textbf{Key Takeaway}\)
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A \(\textbf{sector}\) is a region bounded by two radii and an arc.
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Minor sector is one whose area is less than a half of the area of the circle.
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Major sector is onewhose area is greater than a half of the area of the circle.
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See the figure below;
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The Area of a Sector
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\begin{align*} \text{Area of a Sector}=\amp \frac{\theta}{360} \times \pi r^2 \end{align*}
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where:
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  • \(\theta\) is in degrees,
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  • \(r\) is the radius of the circle,
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  • \(\displaystyle \pi \,≈ \,3.142\, \text{or} \, \frac{22}{7}.\)
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Example 2.6.5.

Find the area of a sector of a circle of radius \(7 \,cm\) if the angle subtended at the centre is \(90^\circ\text{.}\)
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Solution.
The values given are, \(\theta=90^\circ \, , \, r= 7\,cm\)
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\(\text{Area}= \frac{\theta}{360} \times \pi r^2\)
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\begin{align*} \text{Area}=\amp \frac{90}{360} \times \frac{22}{7} \times ( 7^2)\\ =\amp \frac{1}{4} \times \frac{22}{7} \times 49 \\ =\amp \frac{1}{4} \times 22 \times 7\\ =\amp 38.5 \,cm^2 \end{align*}
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Example 2.6.6.

Find the area of a sector of a circle shown below;(use \(\pi=3.142\))
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Solution.
The values given are, \(\theta=45^\circ \, , \, r= 10\,cm\)
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\(\text{Area}= \frac{\theta}{360} \times \pi r^2\)
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\begin{align*} \text{Area}\amp \frac{45}{360} \times 3.142 \times ( 10^2)\\ =\amp \frac{1}{8} \times 3.142 \times 100 \\ =\amp 39.275 \,cm^2 \end{align*}
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Example 2.6.7.

The shaded region in the figure below shows the area swept out on a flat windscreen by a wiper. Calculate the area of this region.
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Solution.
The area of the rigion is goten by subtracting the \(\textbf{Area of the smaller sector}\) from \(\textbf{Area of the larger sector}\) .
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Use \(\text{Area}= \frac{\theta}{360} \times \pi r^2\)
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\(\textbf{Area of the larger sector}\)
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\begin{align*} R= \amp 16\,cm + 4\,cm \\ = \amp 20\,cm \end{align*}
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\begin{align*} \theta= \amp 120^\circ \end{align*}
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\begin{align*} A=\amp \frac{120}{360} \times \frac{22}{7} \times 20^2 \\ =\amp \frac{1}{3} \times \frac{22}{7} \times 400\\ =\amp 419.047619 \,cm^2 \end{align*}
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\(\textbf{Area of the smaller sector}\)
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\begin{align*} r= \amp 16\,cm \end{align*}
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\begin{align*} \theta= \amp 120^\circ \end{align*}
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\begin{align*} A=\amp \frac{120}{360} \times \frac{22}{7} \times 16^2 \\ =\amp \frac{1}{3} \times \frac{22}{7} \times 256\\ =\amp 268.19047 \,cm^2 \end{align*}
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Therefore,
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\begin{align*} \text{Area of the region}=\amp \textbf{Area of the larger sector} -\textbf{Area of the smaller sector}\\ =\amp 419.047619 \,cm^2- 268.19047 \,cm^2\\ =\amp 150.85714\,cm^2 \end{align*}
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\(\textbf{Exercises}\)
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  1. A sector of a circle of radius \(r\)is subtended at the centre by an angle of \(\theta\text{.}\) Calculate the area of the sector if:
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    1. \(\displaystyle r=10\,m ,\quad \theta=264^\circ\)
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    2. \(\displaystyle r=8.4\,cm ,\quad \theta=40^\circ\)
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    3. \(\displaystyle r=1.4\,cm ,\quad \theta=80^\circ\)
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  2. The area of a sector of a circle is \(\,cm^2\) . Find the radius of the circle if the angle subtended at the centre is \(140^\circ\text{.}\) (Take \(\pi= \frac{22}{7}\))
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  3. A goat is tethered at the corner of a fenced rectangular grazing field. If the length of the rope is \(21 \,m\text{,}\) what is its grazing area?
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  4. Shown below is a sector of a circle, with radius \(x\,cm\)
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    The area of the sector is \(18 \pi \,cm^2\)
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    Find the length of \(x\)
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  5. A sector has an angle of \(\frac{\pi}{3}\) radians and a radius of \(8 \,cm\text{.}\) Find its area.
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Checkpoint 2.6.8.

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Checkpoint 2.6.9.

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