The Arc Height (Sagitta) Calculator is a helpful tool for calculating the sagitta, or the height of an arc, based on its radius and the length of the chord. The sagitta is the perpendicular distance from the midpoint of the chord to the arc’s curve. This is an important calculation used in various engineering and architectural fields, particularly when designing structures like bridges, arches, and cable systems, where understanding the height of an arc helps ensure safety, accuracy, and functionality.
Formula
The formula to calculate the Arc Height (Sagitta) is:
s = r ± √(r² – (L² / 4))
Where:
- s is the Sagitta (Arc Height), which is the perpendicular distance from the chord to the arc.
- r is the Radius of the arc.
- L is the Length of the chord.
How to Use
- Enter the Radius (r): Input the radius of the arc, which is the distance from the center of the circle to any point on the arc. This value must be provided in meters.
- Enter the Chord Length (L): Input the length of the straight line (chord) connecting two points on the arc. This value must be in meters.
- Click the “Calculate” Button: After entering both the radius and chord length, click the “Calculate” button.
- View the Result: The calculated sagitta (arc height) will be displayed, which represents the height of the arc above the chord.
Example
Let’s say we have:
- Radius (r) = 5 meters
- Chord Length (L) = 6 meters
Using the formula:
s = 5 ± √(5² – (6² / 4))
s = 5 ± √(25 – 9)
s = 5 ± √16
s = 5 ± 4
So, the arc height (s) can be either:
- s = 5 + 4 = 9 meters, or
- s = 5 – 4 = 1 meter.
Thus, the arc height can be 1 meter or 9 meters depending on the arc’s orientation.
FAQs
- What is sagitta?
Sagitta is the height of an arc from the midpoint of a chord to the arc itself, measured perpendicular to the chord. - Why is sagitta important?
The sagitta is crucial in engineering and construction, as it helps in designing arches, bridges, and determining the shape of curved structures. - What does the formula s = r ± √(r² – (L² / 4)) represent?
This formula calculates the sagitta (arc height) by considering the radius and chord length. The positive or negative root allows for both upward and downward arcs. - How do I measure the radius (r) and chord length (L)?
The radius can be measured from the center of the circle to any point on the arc. The chord length is the straight-line distance between two points on the arc. - What if the radius is smaller than half the chord length?
If the radius is smaller than half the chord length, the square root calculation will result in an error, indicating an invalid input for a real arc. - Can the sagitta be negative?
The sagitta value itself cannot be negative, but the formula accounts for two possible values based on the orientation of the arc. - How can sagitta be used in real-world applications?
Sagitta is used in architecture, civil engineering, and construction for calculating the height of arches and bridges, and in cable systems where understanding the curvature is important. - What does the symbol “±” mean in the formula?
The “±” symbol indicates two possible values for the sagitta: one higher and one lower, depending on the arc’s curvature direction. - Can the arc height be calculated without the radius?
No, the radius is a critical part of the calculation. Without the radius, the sagitta cannot be accurately determined. - What are some practical uses of this calculation?
The sagitta calculation is used in designing bridges, arches, and even cables for suspension systems where accurate curvature measurements are necessary. - Is this formula valid for any arc shape?
Yes, this formula is generally valid for circular arcs, where the arc is part of a perfect circle. - What if the chord length is too long for the given radius?
If the chord length exceeds the allowable limits for a given radius, the formula will return an error, indicating that the inputs do not form a valid arc. - Can this tool be used in designing curved roofs or tunnels?
Yes, engineers use sagitta calculations in the design of curved roofs or tunnels to understand the arc height and the required dimensions for structural integrity. - Does this calculator apply only to physical arcs or also to theoretical ones?
This calculator can be used for both real-world and theoretical arcs, as long as the radius and chord length are known. - What units should be used for the radius and chord length?
The radius and chord length should both be measured in meters to ensure consistency in the calculation and result. - How does changing the radius affect the sagitta?
Increasing the radius generally increases the arc height (sagitta), as the distance from the center of the circle to the arc grows. - Can the chord length ever exceed twice the radius?
No, if the chord length exceeds twice the radius, the arc would no longer form part of a circle, and the formula would no longer apply. - Is there a minimum radius for this calculation?
There is no strict minimum radius, but a very small radius with a large chord length may lead to unrealistic results. - What happens if the inputs are invalid?
If either the radius or chord length is not valid, the tool will display an error or an invalid result message, indicating the problem with the inputs. - Can this calculation be used for elliptical arcs?
No, this specific formula is only for circular arcs. Elliptical arcs require a different set of calculations.
Conclusion
The Arc Height (Sagitta) Calculator is a powerful tool for anyone working with circular arcs in construction, engineering, or design. By knowing the radius and chord length, this calculator allows users to easily compute the height of the arc, ensuring accurate measurements and better-informed decisions when designing structures like arches, bridges, and cables. Understanding the arc height is essential for ensuring safety, stability, and the optimal design of curved systems.