Arch Calculator
Use this arch calculator to find the focus points of an elliptical arch from its span and rise so you can lay out the shape more accurately.
Result
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Quick Answer: For an elliptical arch with span 2a and rise b, the focal distance is c = sqrt(a² - b²). The two focus points are then located at -c and +c from the center line.
How to Calculate
- Measure the full span: Record the width of the opening and divide by two to get the semi-major axis a.
- Measure the rise: Use the vertical height from the spring line to the top of the arch.
- Calculate the focal distance: Apply c = sqrt(a² - b²), where b is the rise of the ellipse used for the arch.
- Mark both focus points: Place the focus points at equal distances left and right of the center line to lay out the elliptical curve.
Formula
Length = 2a; c = sqrt(a² - height²); focus1 = -c; focus2 = +c
| Variable | Meaning | Unit |
|---|---|---|
| a | Semi-major axis or half the span | length |
| height | Arch rise | same as a |
| c | Focal distance from center | same as a |
Worked Examples
Elliptical arch - Omni-style focal-distance example
- Span half-length a: 440 mm
- Rise: 300 mm
Result: Focal distance = 321.9 mm
This matches the worked example described in Omni's FAQ.
Doorway arch - 1.2 m span and 0.4 m rise
- Span: 1.2 m
- Half-span a: 0.6 m
- Rise: 0.4 m
Result: Focal distance = 0.447 m
The focus points sit 0.447 m to either side of the center line.
Shallow arch - Wide opening with lower rise
- Span: 2.0 m
- Half-span a: 1.0 m
- Rise: 0.3 m
Result: Focal distance = 0.954 m
A shallow rise pushes the focus points farther outward along the span.
Interpretation Table
| Range | Meaning | Action |
|---|---|---|
| Higher rise | Focus points move inward | A taller arch tends to reduce focal distance. |
| Lower rise | Focus points move outward | A shallow arch spreads the focus points farther from the center. |
| Rise near half-span | Closer to circular behavior | The ellipse becomes less shallow and easier to distinguish from a wide arch. |
Frequently Asked Questions
Measure the full span and rise, divide the span by two to get a, and then calculate the focal distance with c = sqrt(a² - b²).
Mark the base, center line, and focus points, then use a string or layout method that references both focus points to trace the ellipse.
No. Some arches are semicircular, but elliptical arches follow ellipse geometry rather than circle geometry.
They let you lay out the elliptical curve more accurately on-site or in shop work.
Note: This calculator is for elliptical arch geometry. Other arch types such as semicircular or segmental arches use different geometric rules.
References
Last reviewed: March 14, 2026