The arc radius can be calculated using different formulas depending on the information you have. Here are the primary methods:
1. Using Arc Length and Central Angle (in Radians)
If you know the arc length (s) and the central angle (θ) in radians, the radius (r) is calculated as:
r = s / θ
- s: Arc Length (the distance along the curved line of the arc)
- θ: Central Angle (the angle subtended by the arc at the center of the circle, measured in radians)
- r: Radius of the circle
Example:
Suppose an arc has a length of 10 cm and subtends a central angle of 0.5 radians. Then, the radius would be:
r = 10 cm / 0.5 radians = 20 cm
2. Using Arc Length and Central Angle (in Degrees)
If the central angle (α) is given in degrees, you first need to convert it to radians before using the formula above. The conversion formula is:
*θ = α (π / 180)**
Where:
- α: Central Angle in degrees
- θ: Central Angle in radians
- π: Pi (approximately 3.14159)
Then, use r = s / θ as described above.
Example:
Suppose an arc has a length of 15 cm and subtends a central angle of 60 degrees.
- Convert degrees to radians: θ = 60 * (π / 180) ≈ 1.047 radians
- Calculate the radius: r = 15 cm / 1.047 radians ≈ 14.33 cm
3. Using Chord Length and Sagitta (Height)
If you know the chord length (c) and the sagitta (h - the distance from the midpoint of the chord to the midpoint of the arc), the radius (r) can be calculated using the following formula:
r = (c2 / (8h)) + (h / 2)
- c: Chord Length (the straight-line distance between the two endpoints of the arc)
- h: Sagitta (the height of the arc)
Example:
Suppose an arc has a chord length of 8 cm and a sagitta of 2 cm. Then, the radius would be:
r = (82 / (8 * 2)) + (2 / 2) = (64 / 16) + 1 = 4 + 1 = 5 cm
Summary
The formula to use for calculating the radius of an arc depends on the information readily available. Understanding these formulas allows you to determine the radius with precision, whether you have arc length and central angle data, or chord and sagitta measurements.
