How it works
A chord and its rise (the perpendicular distance from the chord’s midpoint to the arc) fix a unique circle. The chord is half a chord length on each side of the axis, so the radius satisfies (R − h)² + (c/2)² = R², which solves to R = c² ÷ (8h) + h ÷ 2.
The included angle follows from sin(θ/2) = c ÷ (2R), and the arc length is the radius times the angle in radians. Use the radius to strike the curve with a trammel or string for an arch, a curved rafter, or a template.
Worked example
A 96 in span with a 12 in rise:
- Radius = 96² ÷ (8 × 12) + 12 ÷ 2 = 102 in
- Angle = 2 × arcsin(96 ÷ (2 × 102)) = 56.14°
- Arc length = 102 × 56.14° × π ÷ 180 = 99.951 in
| Input | Value |
|---|---|
| Chord length (span) | 96 in |
| Rise (height at the middle) | 12 in |
| Result | Value |
|---|---|
| Radius | 102 in |
| Arc length | 99.951 in |
| Included angle | 56.14 ° |
| Diameter | 204 in |
Assumptions and limits
- A true circular arc. Elliptical and parabolic arches need different geometry.
- Dimensions are measured to the same side of the material (all to the inside edge, for example).
- For layout, mark the center at radius R below the top of the arc, on the perpendicular through the middle of the chord.
Common questions
How do I find the radius of an arch?
Divide the span squared by eight times the rise, then add half the rise.
What is the rise of an arc called?
The sagitta, or versine: the height of the arc above the middle of its chord.
How do I draw the curve?
Find the center at distance R below the top of the arc, on the vertical line through the middle of the chord, and swing a string or trammel of length R.
What if the rise equals half the span?
That is a semicircle: the radius equals the rise.
Sources
- Geometry of a circular segment: R = c² ÷ (8h) + h ÷ 2, from the Pythagorean relation (R − h)² + (c ÷ 2)² = R².
Updated 2026-09-30