Sectors

Arc Length and Sector Area Using Radians:

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$s = r\theta$

Where:

$A = \frac{1}{2}r^2\theta$

Where:

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Example 2: Find the arc length subtended by an angle of $\frac{\pi}{4}$ radians in a circle of radius $10$ cm.

Solution: $s = r\theta = 10 \times \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \text{ cm}$

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Example 3: Calculate the area of a sector with a central angle of $\frac{\pi}{3}$ radians and a radius of $6$ cm.

Solution: $A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times \frac{\pi}{3} = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 18 \times \frac{\pi}{3} = 6\pi \text{ square cm}$

Formula For Degrees:

l = 2\pi r \times \frac{\Theta}{360}

A = \pi r^2 \times \frac{\Theta}{360}

When measured in radians, the formulae are much simpler:

l = r \Theta

A = \frac{1}{2} r^2 \Theta

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Example: Find the area and arc length of the following sector:

l = 8 \times \frac{\pi}{8} = \pi

A = \frac{1}{2} \times (8)^2 \times \frac{\pi}{8} = 4\pi

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Example Question

The diagram shows a sector OAB of a circle, centre $O$ and radius $8$ $cm$ . The angle $AOB$ is $46°$ .

i) Express 46° in radians, correct to 3 significant figures.

\frac{2\pi \text{ radians}}{360^\circ} = \frac{23\pi}{90^\circ} = 46^\circ\\ \equiv 0.803 \text{ radians}

ii) Find the length of the arc AB.

l=r\theta = 8 \times 0.803 = :highlight[6.42 cm] \quad(3sf)

iii) Find the area of the sector OAB.

A=\frac {1}{2}(8)^2 \times \frac {23}{90}\pi = :highlight[25.7cm^2] \quad (3sf)