How this instrument works
A segmental or circular arch — a curved doorway header, a Palladian window top, a bowed transom — is a slice of a larger circle. Span is the straight-line width across the base of the arch, and rise is how far the curve lifts above that base line at its highest point. The arch's radius, the radius of the full circle that curve belongs to, comes from a geometric relationship called the sagitta formula: radius = (rise² + (span/2)²) / (2 × rise).
The sagitta is the technical term for the rise itself — the height of a circular segment measured from the midpoint of its chord to the arc. Given a chord (the span) and a sagitta (the rise), the sagitta formula reconstructs the radius of the circle both belong to, using nothing more than the Pythagorean relationship between a circle's center, its chord, and the arc above it.
Once the radius is known, a framer or mason can strike the actual curve: swing an arc of that radius from a center point positioned below the springline, and the resulting curve passes through both ends of the span and reaches the specified rise at its peak. That is exactly how curved door and window headers, arched fireplace openings, and segmental brick arches get laid out on site, whether with a trammel, a string-and-pencil compass, or a curved template cut from the calculated radius.
A flatter arch — a small rise relative to a wide span — produces a large radius, since it takes a much bigger circle to curve so gently across that width. A deeper, more pronounced arch with a larger rise produces a smaller radius. The formula captures that trade-off automatically: rise sits in the denominator as well as the numerator, so the relationship between the two isn't a simple straight-line scale.
- Enter the arch's full width at the base into Span (ft) — the straight-line distance between the two springing points.
- Enter how far the curve rises above that base line into Rise (ft) — measured at the arch's highest point, the crown.
- Read Arch radius (ft) beneath the inputs — the radius of the circle the arch curve is struck from.
- Rise must be greater than zero; a rise of zero describes a flat lintel, not a curved arch, and has no defined radius under this formula.
Worked example — 20 ft span, 4 ft rise
Enter 20 into Span (ft) and 4 into Rise (ft). The instrument squares the rise (4² = 16) and half the span (10² = 100), adds them to get 116, then divides by twice the rise (2 x 4 = 8): 116 / 8 = 14.5. Arch radius (ft) reads 14.5 ft.
To strike that curve on site, a framer would set a trammel or string to a 14.5 ft radius, position its center on the arch's centerline 10.5 ft below the springline (14.5 ft radius minus the 4 ft rise), and swing an arc between the two springing points 20 ft apart — the resulting curve rises exactly 4 ft above the springline at its center.
Questions
What's the difference between span, rise, and radius?
Span is the straight-line width across the bottom of the arch, between the two points where the curve begins (the springing points). Rise is how high the curve lifts above that base line at its peak, also called the sagitta. Radius is the radius of the full circle that the arch's curve is a slice of — a separate, derived number that a mason or framer needs to actually strike the curve with a trammel or template, since neither span nor rise alone describes the circle.
Why does a flatter arch have a larger radius?
Because it takes a much bigger circle to curve gently across a wide span with only a small rise. As rise shrinks toward zero relative to span, the required radius grows toward infinity — in the limit, a perfectly flat lintel behaves like an arc of an infinitely large circle. A deeply curved arch with a large rise, by contrast, comes from a much smaller, tighter circle.
How do I use the radius to actually mark the curve?
Find the center point: on the vertical centerline of the span, measure down from the springline (the base) by radius minus rise. Anchor a trammel, string compass, or router jig at that center and swing an arc of the calculated radius — it will pass through both springing points and reach the specified rise at the crown, giving the exact curve to cut or lay masonry to.
What happens if I enter a rise of zero?
The instrument won't compute a radius, because dividing by twice the rise means dividing by zero when rise is zero — and geometrically, a rise of zero describes a flat lintel rather than a curved arch, which has no meaningful radius under the sagitta formula. Enter a rise greater than zero for any true curved arch.
Does this formula work for a Gothic or pointed arch?
No — it assumes a single-radius circular (segmental) arch, where the whole curve is struck from one center point. A Gothic or pointed arch is typically built from two intersecting arcs with two different centers, and elliptical or parabolic arches follow a different curve family entirely, so this sagitta formula does not apply to those shapes.
Is 'rise' the same measurement as 'sagitta'?
Yes — sagitta is the formal geometric term for exactly the measurement carpenters and masons call the rise: the height of a circular segment from the midpoint of its chord (the span) up to the arc. The two words describe the same distance in the same formula, one from classical geometry and one from the trade.