The arc length of a circle is found using a formula that relates the radius of the circle and the central angle subtended by the arc.
Here's a breakdown of how to calculate arc length:
Understanding Arc Length
Arc length refers to the distance along the curved line of an arc, which is a portion of a circle's circumference.
Formula for Arc Length
There are two common ways to calculate arc length, depending on whether the central angle is given in degrees or radians:
-
When the central angle (θ) is in degrees:
Arc Length (s) = (θ/360) * 2πr
Where:
- s = arc length
- θ = central angle in degrees
- r = radius of the circle
- π (pi) ≈ 3.14159
-
When the central angle (θ) is in radians:
Arc Length (s) = rθ
Where:
- s = arc length
- θ = central angle in radians
- r = radius of the circle
Explanation of the Formulas
- The formula using degrees essentially calculates what fraction of the entire circumference (2πr) the arc represents. (θ/360) gives you that fraction.
- The formula using radians directly relates the radius and the angle. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius.
Example
Let's say you have a circle with a radius of 5 cm, and you want to find the arc length corresponding to a central angle of 60 degrees.
Using the degree formula:
s = (60/360) 2π(5)
s = (1/6) 10π
s = (5/3)π cm
s ≈ 5.24 cm
Therefore, the arc length is approximately 5.24 cm.
In summary, you find arc length by determining the fraction of the circle's circumference that the arc represents, based on its central angle, and then multiplying that fraction by the circumference (or directly multiplying the radius by the angle in radians).
