Geometry and Trigonometry - Radian measure (HL)

A sector of a circle has a radius of $8 \text{ cm}$ and an arc length of $12 \text{ cm}$. Find the angle $\theta$ in radians and the area of the sector.

$s = r\theta \implies 12 = 8\theta \implies \theta = 1.5 \text{ rad}$ $A = \frac{1}{2}r^2\theta = \frac{1}{2}(8^2)(1.5) = \frac{1}{2}(64)(1.5) = 48 \text{ cm}^2$

First, use the arc length formula $s = r\theta$ to solve for $\theta$. Then, substitute the radius and the calculated angle into the sector area formula.

Convert $210^\circ$ into radians, leaving your answer in terms of $\pi$.

$\text{Angle in radians} = 210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}$

To convert degrees to radians, multiply the degree value by $\frac{\pi}{180}$ and simplify the fraction.

A circle has a radius of $10 \text{ cm}$. A chord subtends an angle of $1.2 \text{ rad}$ at the center. Calculate the area of the minor segment.

$\text{Area} = \frac{1}{2}r^2(\theta - \sin\theta)$ $\text{Area} = \frac{1}{2}(10^2)(1.2 - \sin(1.2))$ $\text{Area} = 50(1.2 - 0.93203...)$ $\text{Area} \approx 13.4 \text{ cm}^2$

The area of a segment is found by subtracting the triangle area $\frac{1}{2}r^2\sin\theta$ from the sector area $\frac{1}{2}r^2\theta$. Ensure the calculator is in Radian mode when calculating $\sin(1.2)$.

$r = 40$ $\theta = \frac{2\pi}{3}$ $A = \frac{1}{2}r^2\theta$ $A = \frac{1}{2} \times 40^2 \times \frac{2\pi}{3}$ $A = 800 \times \frac{2\pi}{3} = \frac{1600\pi}{3}$ $A \approx 1675.5 \text{ cm}^2$

Identify the radius as the length of the wiper and the angle as the sweep. Use the sector area formula $A = \frac{1}{2}r^2\theta$ because the angle is already in radians.

$\text{Perimeter} = s + 2r = 9 + 2(6) = 21 \text{ cm}$ $\theta = \frac{s}{r} = \frac{9}{6} = 1.5 \text{ rad}$ $\text{Chord length} = 2r\sin\left(\frac{\theta}{2}\right)$ $\text{Chord length} = 2(6)\sin(0.75)$ $\text{Chord length} \approx 12 \times 0.6816 = 8.18 \text{ cm}$

The perimeter is the sum of the arc length and the two bounding radii. To find the chord length, first determine the central angle $\theta$ in radians, then apply the chord formula.