The Arc Length Formula: How to Find the Length of an Arc on a Circle with the Radius and Central Angle

Arc Length Formula

The arc length formula is used to find the length of an arc on a circle, given the radius and the central angle

The arc length formula is used to find the length of an arc on a circle, given the radius and the central angle.

The formula is:

Arc Length = (Central angle in degrees / 360) x (2πr)

Where:
– Arc Length is the length of the arc.
– Central angle in degrees is the measure of the angle formed by the two radii that make up the arc, with the center of the circle as the vertex of the angle.
– π (pi) is a mathematical constant approximately equal to 3.14159.
– r is the radius of the circle.

To use the formula, follow these steps:

1. Convert the central angle to degrees if it is given in radians. One full revolution or circle is equal to 360 degrees or 2π radians. So if the angle is given in radians, multiply it by (180/π) to convert it to degrees.

2. Plug the central angle in degrees into the formula.

3. Multiply the central angle in degrees divided by 360 by (2πr).

4. Simplify the expression if possible.

Here’s an example to help illustrate the use of the arc length formula:

Let’s say we have a circle with a radius of 5 cm and a central angle of 120 degrees. We want to find the length of the corresponding arc.

Arc Length = (Central angle / 360) x (2πr)
= (120 degrees / 360) x (2π x 5 cm)
= (1/3) x (2π x 5 cm)
= (1/3) x (10π cm)
= (10/3)π cm

So the length of the arc is approximately (10/3)π cm, or approximately 10.47 cm.

Remember to always check whether the central angle is given in degrees or radians, and adjust your calculations accordingly.

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