Minor arc

Definition

A minor arc is a portion of a circle defined by two endpoints on the circle, representing the shorter path between those points along the circumference. Minor arcs are significant in understanding relationships within circles, as they help define angles, segments, and chords that contribute to the overall geometry of the circle.

5 Must Know Facts For Your Next Test

1. Minor arcs are always less than 180 degrees in measurement.
2. The length of a minor arc can be calculated using the formula: $$L = \frac{\theta}{360} \times 2\pi r$$ where $$\theta$$ is the central angle in degrees and $$r$$ is the radius of the circle.
3. Two minor arcs can be formed from any two points on a circle, but only one of them will be shorter and hence classified as a minor arc.
4. The measure of a minor arc is equivalent to the measure of its corresponding central angle.
5. In a circle, the minor arc can be labeled with its endpoints, typically denoted as $$AB$$ for a minor arc between points A and B.

Review Questions

• How do you determine whether an arc is classified as a minor arc or a major arc?
  • To determine if an arc is a minor or major arc, you look at its measurement in degrees. A minor arc measures less than 180 degrees, making it the shorter path between two endpoints on the circle. Conversely, if an arc exceeds 180 degrees, it is classified as a major arc. Understanding this distinction helps in solving problems related to angles and lengths in circles.
• Explain how to calculate the length of a minor arc given its central angle and radius.
  • To calculate the length of a minor arc, use the formula: $$L = \frac{\theta}{360} \times 2\pi r$$. Here, $$\theta$$ represents the central angle measured in degrees, and $$r$$ is the radius of the circle. This formula takes into account that the full circumference of the circle is $$2\pi r$$ and proportionally calculates how much of that circumference corresponds to the minor arc defined by the given central angle.
• Discuss how minor arcs relate to other elements of circles, such as chords and central angles, and why these relationships are important in geometry.
  • Minor arcs are closely related to chords and central angles within circles. The endpoints of a chord define a minor arc, and this relationship allows for a deeper understanding of circle properties. The measure of a minor arc directly corresponds to its associated central angle, reinforcing connections between angles and arcs. These relationships are vital in solving complex geometric problems and in applications like navigation and architecture, where precise measurements are essential.

Related terms

major arc:

A major arc is the longer path between two endpoints on a circle, covering more than half of the circle's circumference.

chord: A chord is a line segment with both endpoints on the circle, which may define a minor arc between those endpoints.

central angle:

A central angle is an angle whose vertex is at the center of the circle and whose sides intersect the circle, often helping to define minor arcs.