Angles, Radian Measure, Arc Length, and Speed in Circles

Angles and Their Measure

Degree Measure

Angles are fundamental geometric objects formed by two rays sharing a common endpoint, called the vertex. The measure of an angle is typically expressed in degrees or radians.

• Angle: Two rays with a common endpoint (vertex).

• Central Angle: An angle whose vertex is at the center of a circle.

Degrees of Some Common Central Angles:

• 0° (straight right)

• 90° (upward)

• 180° (left)

• 270° (downward)

• 360° (full rotation)

Counterclockwise angles are positive. Clockwise angles are negative.

Example: Draw angles of 45°, -270°, 225°, and 405°.

Radian Measure

Degrees and Radians

Angles can also be measured in radians, which relate the angle to the radius of a circle.

• 360° = $2\pi$ radians

• 180° = $\pi$ radians

To convert from degrees to radians, multiply by $\frac{\pi}{180}$.

To convert from radians to degrees, multiply by $\frac{180}{\pi}$.

Examples:

• Convert 90° to radians: $90° \times \frac{\pi}{180} = \frac{\pi}{2}$ radians

• Convert 150° to radians: $150° \times \frac{\pi}{180} = \frac{5\pi}{6}$ radians

• Convert $\frac{\pi}{4}$ radians to degrees: $\frac{\pi}{4} \times \frac{180}{\pi} = 45°$

• Convert $-\frac{3\pi}{2}$ radians to degrees: $-\frac{3\pi}{2} \times \frac{180}{\pi} = -270°$

Dimensional Analysis

Dimensional analysis is used to convert units, such as yards to centimeters.

• Example: Convert 6.5 yd to cm. $6.5 \text{ yd} \times 3 \text{ ft/yd} \times 12 \text{ in/ft} \times 2.54 \text{ cm/in}$

Arc Length

Definition and Formula

The arc length of a complete circle is equal to its circumference. For a given radius, the arc length depends on the measure of the central angle.

• Arc Length (degrees): If a central angle of $\theta$ degrees intercepts an arc in a circle of radius $r$, then $s = \frac{\theta}{360} \times 2\pi r$

• Arc Length (radians): If a central angle of $\theta$ radians intercepts an arc in a circle of radius $r$, then $s = r\theta$

Example: Find the arc length for a circle of radius 2 m subtended by a central angle of $\theta = 2\pi/3$ radians: $s = 2 \times \frac{2\pi}{3} = \frac{4\pi}{3}$ m

Example: Find the arc length for a circle of radius 3 m subtended by a central angle of 160°: $s = 3 \times \frac{160}{360} \times 2\pi = 3 \times \frac{4}{9} \pi \approx 4.19$ m

Area of a Sector of a Circle

Definition and Formula

The area of a sector is the region bounded by two radii and the arc they intercept. It depends on the central angle and the radius.

• Area (degrees): $A = \frac{\theta}{360} \times \pi r^2$

• Area (radians): $A = \frac{1}{2} r^2 \theta$

Example: Find the area of a sector for a circle with $r = 2$ ft and $\theta = 3\pi/4$ radians: $A = \frac{1}{2} \times 2^2 \times \frac{3\pi}{4} = 2 \times \frac{3\pi}{4} = \frac{3\pi}{2}$ ft2

Example: Find the area of a sector for a circle with $r = 6$ inches and central angle 45°: $A = \frac{45}{360} \times \pi \times 36 = \frac{1}{8} \times \pi \times 36 = 4.5\pi$ in2

Linear and Angular Speed

Linear Speed

Linear speed is the rate at which an object moves along a circular path.

• Formula: $v = \frac{s}{t}$, where $s$ is arc length and $t$ is time.

• Alternatively, $v = r\omega$, where $r$ is radius and $\omega$ is angular speed.

Angular Speed

Angular speed measures how quickly the central angle changes as an object moves around a circle.

• Formula: $\omega = \frac{\theta}{t}$, where $\theta$ is the angle in radians and $t$ is time.

Example: If an object travels around a circle of radius 6 ft and sweeps out 1.7 radians in 60 seconds, $\omega = \frac{1.7}{60} \approx 0.028$ radians/sec.

Example: Find the angular speed of an engine idling at 900 rpm: $\omega = 900 \times \frac{2\pi}{60} = 30\pi$ radians/minute

Converting Angular Speed to Linear Speed

To convert angular speed to linear speed, use $v = r\omega$.

• Example: A child spins a rock at the end of a 2-ft rope at 180 rpm. Find the linear speed when released: $\omega = 180 \times \frac{2\pi}{60} = 6\pi$ radians/sec, $v = 2 \times 6\pi = 12\pi$ ft/sec