Arc

Definition

An arc is a portion of the circumference of a circle. It is a curved line segment that connects two points on the circle's perimeter, representing an angular measure between those points.

5 Must Know Facts For Your Next Test

1. The measure of an arc is expressed in degrees or radians, and it represents the fraction of the total circumference of the circle.
2. The length of an arc is proportional to the measure of the central angle that subtends it.
3. Arcs can be classified as major arcs (greater than 180 degrees) or minor arcs (less than 180 degrees).
4. The formula for the length of an arc is: $L = r\theta$, where $L$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the measure of the central angle in radians.
5. Arcs are used in the calculation of sector area and segment area within a circle.

Review Questions

• Explain the relationship between the measure of a central angle and the length of the corresponding arc.
  • The measure of a central angle and the length of the corresponding arc are directly proportional. As the central angle increases, the length of the arc also increases proportionally. This relationship is expressed by the formula $L = r\theta$, where $L$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the measure of the central angle in radians. The larger the central angle, the greater the fraction of the circle's circumference that the arc represents.
• Describe how arcs are used in the calculation of sector area and segment area within a circle.
  • Arcs play a crucial role in the calculation of sector area and segment area within a circle. The sector area is the area bounded by an arc and the two radii that intersect at the endpoints of the arc. The formula for sector area is $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the measure of the central angle in radians. The segment area is the area bounded by a chord and the arc it subtends. The formula for segment area involves the arc length and the radius of the circle, and it can be used to find the area of irregular figures inscribed within a circle.
• Analyze how the classification of arcs as major or minor can impact the calculations and interpretations within geometric applications.
  • The classification of arcs as major or minor is important in geometric applications. Major arcs are greater than 180 degrees, while minor arcs are less than 180 degrees. This distinction can affect the calculations and interpretations within circle problems. For example, when finding the length of an arc, the formula $L = r\theta$ applies to both major and minor arcs, but the value of $\theta$ must be adjusted accordingly. Additionally, the classification of arcs can impact the determination of sector area and segment area, as the formulas and approaches may differ depending on whether the arc is a major or minor arc.

Related terms

Chord: A straight line segment that connects two points on the circumference of a circle.

Central Angle: The angle formed by two radii of a circle that intersect at the center, and the measure of the arc between those radii.

Inscribed Angle: An angle formed by two chords that intersect on the circumference of a circle.