To find the length of a minor arc associated with a chord in a circle, you need to know the radius of the circle and the measure of the central angle that subtends the arc.
A chord is a line segment connecting two points on a circle. Every chord divides a circle into two arcs: a minor arc and a major arc. The minor arc is the shorter of the two. The length of this minor arc is directly related to the central angle formed by connecting the endpoints of the chord to the center of the circle.
The Formula for Minor Arc Length
The length of a minor arc is calculated using the radius of the circle and the measure of the central angle (θ) subtended by the arc. Based on the provided information, the formula to find the length of the minor arc is:
Arc Length = (π r θ) / 180
Where:
- π (Pi) is a mathematical constant, approximately 3.14159.
- r is the radius of the circle.
- θ is the measure of the central angle subtended by the minor arc, measured in degrees.
This formula essentially calculates what fraction of the circle's circumference the arc represents (determined by θ/360) and then multiplies that fraction by the total circumference (2πr). The formula (π * r * θ) / 180 is derived from (θ/360) * 2πr by simplifying.
Steps to Calculate Minor Arc Length
Finding the length of a minor arc is straightforward if you have the necessary information.
Here are the steps:
- Identify the Radius (r): Determine the radius of the circle.
- Find the Central Angle (θ): Determine the measure of the central angle that corresponds to the minor arc. This is the angle formed at the center of the circle by lines extending to the two endpoints of the chord. Ensure this angle is in degrees.
- Apply the Formula: Substitute the values of
randθinto the formula:Arc Length = (π * r * θ) / 180. - Calculate: Perform the calculation to find the arc length.
Example Calculation
Let's calculate the length of a minor arc with a central angle of 60 degrees in a circle with a radius of 10 cm.
- Radius (r) = 10 cm
- Central Angle (θ) = 60 degrees
Using the formula:
Arc Length = (π r θ) / 180
Arc Length = (π 10 60) / 180
Arc Length = (600π) / 180
Arc Length = (600/180) π
Arc Length = (10/3) π cm
So, the length of the minor arc is (10π/3) cm, which is approximately 10.47 cm.
Key Elements for Calculation
| Element | Symbol | Description | Unit (for formula) |
|---|---|---|---|
| Radius | r | Distance from the center to any point on the circle | Units of length |
| Central Angle | θ | Angle subtended by the arc at the circle's center | Degrees (° ) |
| Pi | π | Mathematical Constant (approx. 3.14159) | None |
Knowing the radius and the central angle are crucial for using the arc length formula effectively. If you only have the chord length and radius, you would first need to calculate the central angle using trigonometry before finding the arc length.
