What is arc length divided by circumference?

Arc length divided by the circumference of a circle is equal to the ratio of the central angle subtended by the arc to 360 degrees.

Explanation

The circumference of a circle represents the total distance around the circle, corresponding to a central angle of 360 degrees. An arc length is a portion of that circumference, corresponding to a specific central angle. Therefore, the ratio of the arc length to the circumference will always be equivalent to the ratio of the arc's central angle to the full circle's angle (360°).

Mathematically, this can be represented as follows:

• Let s be the arc length.
• Let C be the circumference of the circle.
• Let θ (theta) be the central angle subtended by the arc (in degrees).

Then, the relationship is:

s / C = θ / 360°

Example

Suppose an arc has a central angle of 90 degrees in a circle.

The ratio of the arc length to the circumference is:

90° / 360° = 1/4

This means that the arc length is one-quarter of the circle's circumference.

Significance

This relationship is fundamental in understanding the geometry of circles and arcs. It allows for the calculation of arc lengths if the central angle and circumference are known, or vice-versa. It provides a proportional relationship between the arc length and the central angle, making it a valuable tool in various mathematical and practical applications.