The arc length of a circle is the distance along a portion of its circumference, calculated by using the circle's radius and the central angle that defines that specific arc.
Understanding Arc Length
An arc is essentially a segment of a circle's circumference. To determine its length, you need two fundamental pieces of information:
- Radius (r): The distance from the center of the circle to any point on its circumference.
- Central Angle (θ): The angle formed by two radii connecting the center to the endpoints of the arc. This angle is crucial and can be measured in either radians or degrees, dictating which formula you should use.
Arc Length Formulas
The method for calculating arc length depends on the unit of measurement used for the central angle (θ).
When the Central Angle is in Radians
If the central angle (θ) is measured in radians, the formula for arc length (L) is straightforward:
L = θ × r
- Where L is the arc length.
- θ (theta) is the central angle in radians.
- r is the radius of the circle.
When the Central Angle is in Degrees
If the central angle (θ) is measured in degrees, you need to convert the degrees to radians as part of the calculation, or use a specific formula that incorporates this conversion:
L = θ × (π/180) × r
- Where L is the arc length.
- θ (theta) is the central angle in degrees.
- π (pi) is a mathematical constant approximately equal to 3.14159.
- r is the radius of the circle.
Summary of Formulas
For quick reference, here's a table summarizing the arc length formulas:
| Angle Unit | Formula for Arc Length (L) |
|---|---|
| Radians | L = θ × r |
| Degrees | L = θ × (π/180) × r |
Practical Examples
To illustrate how these formulas are applied:
-
Example 1 (Using Radians):
- Problem: Find the arc length of a circle with a radius of 10 cm and a central angle of π/2 radians.
- Solution: Using L = θ × r, we get L = (π/2) × 10 = 5π cm.
-
Example 2 (Using Degrees):
- Problem: Calculate the arc length of a circle with a radius of 7 meters and a central angle of 90 degrees.
- Solution: Using L = θ × (π/180) × r, we get L = 90 × (π/180) × 7.
- This simplifies to L = (1/2) × π × 7 = 3.5π meters.
Understanding these formulas allows you to accurately determine the length of any arc given the circle's radius and its corresponding central angle.
