Arc height, also known as sagitta, measures the distance from the midpoint of a chord to the peak of the arc. This calculator determines both the small arc height (sagitta) and large arc height for any circular arc.
Formula
The arc height is calculated using:
[s = r \pm \sqrt{r^2 - \left(\frac{c}{2}\right)^2}]
Where:
- s is the arc height
- r is the radius of the circle
- c is the chord length
The formula gives two values:
- Small arc height (sagitta): (s = r - \sqrt{r^2 - \left(\frac{c}{2}\right)^2})
- Large arc height: (s = r + \sqrt{r^2 - \left(\frac{c}{2}\right)^2})
Calculation Example
For an arc with radius 8 inches and chord length 10 inches:
[s = 8 \pm \sqrt{8^2 - \left(\frac{10}{2}\right)^2}]
[s = 8 \pm \sqrt{64 - 25}]
[s = 8 \pm \sqrt{39}]
[s \approx 8 \pm 6.245]
Results:
- Large Arc Height: 14.245 inches
- Small Arc Height (Sagitta): 1.755 inches
Applications
| Field | Use Case |
|---|---|
| Architecture | Designing arches and domes |
| Engineering | Bridge construction |
| Optics | Lens curvature calculations |
| Manufacturing | Creating curved surfaces |
Important Constraints
- The chord length cannot exceed the diameter (2 × radius)
- Both values must be positive numbers
- The small arc height is always less than or equal to the radius