What this calculator does
A circular arch is defined by its span and rise, and the radius follows from the sagitta formula. A 900 mm span with a 150 mm rise needs a radius of 750 mm.
The relationship is not intuitive, because a shallow arch needs a very large radius. Halving the rise to 75 mm nearly doubles the radius to 1,387.5 mm, and a very flat arch quickly needs a radius larger than the room it is being built in. That is why flat arches are set out with a trammel or by calculation rather than by swinging a compass.
The formula
The radius is the rise divided by two plus the span squared over eight times the rise, which is the standard circular segment relationship. The included angle follows from the geometry, and the arc length is the curve you actually cut or bend. Where the rise equals half the span, the arch is a semicircle and the angle reaches 180 degrees.
| Term | Meaning |
|---|---|
| Span | The width of the arch opening, measured across the springing points. |
| Rise | The height of the arch above the line joining the springing points. |
| Sagitta | The rise, in the geometry of a circular segment. Latin for arrow. |
| Springing point | Where the arch meets its supports, at each end of the span. |
The inputs explained
| Field | What to enter |
|---|---|
| Span (arch width) (mm) | Arch width in millimetres. |
| Rise (arch height) (mm) | Arch height above the springing line. Half the span gives a semicircle. |
When to use it
Setting out an arched opening
The radius is what you need to strike the curve with a trammel or string.
Cutting an arched head
The arc length is the curve to cut, which is longer than the span.
Bending trim to an arch
Flexible moulding is ordered by the arc length, not by the span.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the rise change the radius?
The same span at a range of arch rises.
| Rise | Arc radius | Included angle | Arc length (curve to cut) |
|---|---|---|---|
| 75 mm | 1,387.5 mm | 37.8° | 916.6 mm |
| 150 mm | 750.0 mm | 73.7° | 965.3 mm |
| 300 mm | 487.5 mm | 134.8° | 1,146.6 mm |
| 450 mm | 450.0 mm | 180.0° | 1,413.7 mm |
Questions
Why does a flat arch need such a large radius?
Because a shallow curve is a small piece of a very big circle. As the rise approaches zero the radius approaches infinity, which is the straight line. This is why flat arches cannot be struck with a compass and are set out by calculation or with a long trammel.
What is a segmental arch?
Any arch that is less than a semicircle, which is every case here except the 450 mm rise. It takes a segment of a circle rather than a half, and it is the most common form for openings because it needs less height above the opening.
How do I strike the curve on site?
From the radius point, which for a shallow arch sits well below the opening, often below floor level. A trammel bar pivoting on a fixed point is the usual method. For very flat arches a string line from a stake outside the building is sometimes the only option.
Why is the arc length longer than the span?
Because it follows the curve rather than the straight line across. A 900 mm span with a 150 mm rise has an arc of 965.3 mm. Order trim to the arc length, not the span, or you will be short.
For angled cuts, see the mitre cut calculator. For roof geometry, see the roof pitch calculator.