To find the arc length of a circle, you'll need to know the circle's radius and the central angle subtended by the arc. The formula used varies depending on whether the angle is given in radians or degrees. Here's a breakdown:
Understanding Arc Length
Arc length refers to the distance along the curved edge of a circle’s section, known as an arc. It's a portion of the entire circumference.
Formulas for Arc Length
The following formulas, based on the provided references, enable us to calculate arc length:
| Angle Measurement | Formula |
|---|---|
| Radians | Arc Length = θ × r |
| Degrees | Arc Length = θ × (π/180) × r |
Where:
- θ (theta) is the central angle.
- r is the radius of the circle.
Steps to Calculate Arc Length
- Identify the given values: Determine the radius (r) of the circle and the central angle (θ).
- Check the unit of the angle:
- Is the angle given in radians or degrees?
- Select the correct formula:
- Use Arc Length = θ × r if the angle is in radians.
- Use Arc Length = θ × (π/180) × r if the angle is in degrees.
- Plug in the values: Substitute the known values into the chosen formula.
- Calculate: Perform the multiplication to find the arc length. The unit of the arc length will be the same as the unit of the radius.
Example Calculation
Let's illustrate with examples:
-
Example 1 (Radians):
- Suppose a circle has a radius (r) of 5 cm.
- The central angle (θ) is 1.2 radians.
- Arc Length = 1.2 × 5 = 6 cm
-
Example 2 (Degrees):
- Suppose a circle has a radius (r) of 10 inches.
- The central angle (θ) is 60 degrees.
- Arc Length = 60 × (π/180) × 10 ≈ 10.47 inches
Important Notes:
- Ensure your calculator is set to the correct mode (radians or degrees) before computing if you are not performing the calculations manually.
- The formula Arc Length = θ × r applies only when θ is measured in radians.
- The radian measure of an angle is the ratio of the arc length to the radius of the circle.
By following these steps and formulas, you can accurately calculate the arc length for any given circle and angle.
