To find the length of a segment in a circle, you need to understand what a segment is. A segment is the area of a circle that is ‘cut off’ from the rest of the circle by a chord. The length of the segment can be determined using a formula that involves the radius of the circle and the angle subtended by the chord at the center of the circle.
The basic formula to find the length of the segment, denoted as S, is:
S = r^2 (θ – sin(θ))
Where:
- r = radius of the circle
- θ = central angle in radians that subtends the chord
To apply this formula, follow these steps:
- Determine the radius of the circle (r).
- Calculate the central angle (θ) in radians. If the angle is given in degrees, convert it to radians by using the formula: θ = degrees × (π / 180).
- Plug the values into the formula to find the length of the segment.
For example, if you have a circle with a radius of 10 units and the angle subtended is 60 degrees:
- Convert 60 degrees to radians: θ = 60 × (π / 180) = π/3.
- Now apply the formula: S = 10^2 (π/3 – sin(π/3)).
- Calculate the result for S to find the length of the segment.
This method allows you to find the length of the segment efficiently. Remember that understanding the geometry of circles is key to applying this formula correctly.