Central angle

Definition

A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to the circumference of the circle. This type of angle helps define sectors of the circle and relates to arc lengths, making it a foundational concept in understanding circles and their properties.

5 Must Know Facts For Your Next Test

1. The measure of a central angle in degrees is equal to the measure of the arc it intercepts.
2. Central angles can be measured in both degrees and radians, with a full circle corresponding to $360$ degrees or $2\pi$ radians.
3. In a circle, there can be infinitely many central angles, but they will always add up to $360$ degrees when combined.
4. Central angles help calculate important aspects of circles, such as the area of a sector and the length of an arc.
5. When two central angles have the same measure, they intercept arcs that are also equal in length.

Review Questions

• How does a central angle relate to the arc it intercepts within a circle?
  • A central angle directly determines the measure of the arc it intercepts. Specifically, the degree measure of the central angle is equal to the degree measure of its intercepted arc. This relationship allows for calculations involving arc lengths and areas of sectors based on known central angles, illustrating how central angles play a crucial role in circular geometry.
• What are the differences between central angles and inscribed angles in terms of their measures and properties?
  • Central angles and inscribed angles differ in how they are defined and their measures relative to arcs. A central angle has its vertex at the center of the circle and its measure equals that of the arc it intercepts. In contrast, an inscribed angle has its vertex on the circumference and measures half that of its intercepted arc. This fundamental difference highlights how these angles interact with circular geometry differently.
• Evaluate how knowledge of central angles contributes to understanding advanced concepts like sector area and arc length calculations in circular geometry.
  • Understanding central angles is essential for calculating sector areas and arc lengths in circular geometry. The formula for sector area involves the central angle's measure: Area = $rac{\theta}{360} \cdot \pi r^2$ for degrees or Area = $rac{1}{2} r^2 \theta$ for radians. Similarly, arc length can be determined using the relation: Arc Length = $rac{\theta}{360} \cdot 2\pi r$ for degrees or Arc Length = $r\theta$ for radians. This deepens comprehension of circles and showcases the practical applications of central angles in problem-solving.

Related terms

circumference: The total distance around the edge of a circle, calculated as $C = 2\pi r$, where $r$ is the radius.

arc: A part of the circumference of a circle, defined by two endpoints on the circle, which is directly related to the measure of the central angle.

sector: A region of a circle enclosed by two radii and the arc between them, effectively representing the 'slice' of the circle that corresponds to a central angle.