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° = radians

• 180° = radians

To convert from degrees to radians, multiply by .

To convert from radians to degrees, multiply by .

Examples:

• Convert 90° to radians: radians

• Convert 150° to radians: radians

• Convert radians to degrees:

• Convert radians to degrees:

Dimensional Analysis

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

• Example: Convert 6.5 yd to cm.

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 degrees intercepts an arc in a circle of radius , then

• Arc Length (radians): If a central angle of radians intercepts an arc in a circle of radius , then

Example: Find the arc length for a circle of radius 2 m subtended by a central angle of radians: m

Example: Find the arc length for a circle of radius 3 m subtended by a central angle of 160°: 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):

• Area (radians):

Example: Find the area of a sector for a circle with ft and radians: ft2

Example: Find the area of a sector for a circle with inches and central angle 45°: in2

Linear and Angular Speed

Linear Speed

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

• Formula: , where is arc length and is time.

• Alternatively, , where is radius and is angular speed.

Angular Speed

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

• Formula: , where is the angle in radians and is time.

Example: If an object travels around a circle of radius 6 ft and sweeps out 1.7 radians in 60 seconds, radians/sec.

Example: Find the angular speed of an engine idling at 900 rpm: radians/minute

Converting Angular Speed to Linear Speed

To convert angular speed to linear speed, use .

• Example: A child spins a rock at the end of a 2-ft rope at 180 rpm. Find the linear speed when released: radians/sec, ft/sec