Arch Layout Calculator
Find the radius and arc length for a curved arch, from its span and rise.
Reviewed by the ToolNestr Editorial Team — July 2026
How the arch layout calculator works
Laying out a curved arch — over a doorway, window, or piece of trim — comes down to finding the radius of the circle that the arc is a slice of. Given the span (the straight-line width of the opening at its base, called the springline) and the rise (the height of the curve above that springline), the radius is found with R = (rise² + (span/2)²) ÷ (2 × rise). This formula comes from the geometric relationship between a chord, its height above the chord, and the radius of the circle it belongs to.
Once you have the radius, you can find the arc length — the actual curved distance along the top of the opening — which tells you how much flexible trim, casing, or track material to buy. Arc length equals the radius times the sweep angle in radians; the calculator works out that angle from the span and radius and converts the result to a length automatically.
To physically mark the curve for cutting, use a trammel — essentially a large compass — pivoted at the arc's true center. That center sits directly below the midpoint of the span, at a distance equal to the radius measured down from the highest point of the arc; the calculator reports how far that center point falls below the springline, which is what you'd measure down from the base of the opening to locate the pivot point on the wall or a temporary layout batten.
Arc geometry (chord, rise, radius relationship) follows standard circular-segment formulas used in finish carpentry and millwork trade references.
Set span and rise
Width of the opening and how tall the curve is.
Get the radius
Used to strike the curve with a trammel.
Order trim by arc length
The actual curved distance to cover.
The formula explained
Radius
= (rise² + (span⁄2)²) ÷ (2×rise). A 48 in span, 6 in rise: (36+576) ÷ 12 = 51 in.
Sweep angle
= 2 × asin((span⁄2) ÷ R) = 2 × asin(24÷51) = 56.1°.
Arc length
= R × angle(rad) = 51 × 0.980 = 50.0 in.
Worked example
A wide 72 in doorway arch with a 12 in rise.
Types of arches this calculator handles
Full semicircle arches
When the rise equals half the span, the arc is a true semicircle and the radius equals the half-span exactly — a common look for formal entryways and Palladian-style windows.
Segmental arches
A shallow rise relative to a wide span produces a large radius and a gentle curve — common over standard door and window openings where headroom is limited.
Trammel layout
Larger radii push the trammel's pivot point far below the springline, sometimes off the wall entirely — for those, a shop-made trammel arm or a story pole with multiple points approximates the curve.
Trim and casing stock
Flexible arch casing is typically sold by the linear foot; use the calculated arc length, rounded up, to know how much to order and avoid a mid-project shortfall.
Real-world context
A true semicircular entry arch is the cleanest case. A 36-inch-wide opening with an 18-inch rise — where rise equals exactly half the span — gives R = (18² + 18²) ÷ (2×18) = 18 inches, matching the half-span precisely, with a full 180° sweep and an arc length of 56.55 inches, since a true semicircle is really just a special case of the general formula.
A shallow segmental arch over a standard door shows how flat curves behave. A wide 60-inch opening with just a 4-inch rise produces R = (16 + 900) ÷ 8 = 114.5 inches — a large, gentle radius with only a 30.4° sweep and a 60.71-inch arc length, and a trammel center that falls 110.5 inches below the springline, often too far to swing from inside the room without an extended beam.
A moderate window arch splits the difference. A 30-inch span with a 5-inch rise gives R = (25 + 225) ÷ 10 = 25 inches, a 73.7° sweep, and a 32.18-inch arc length, with the trammel center 20 inches below the springline — a radius small enough to strike with a standard shop-made trammel arm pinned at the wall.
Common misconceptions
"A bigger rise always means a bigger radius." It's the opposite for a fixed span. As rise increases toward half the span, the radius actually shrinks toward the half-span value; small rises relative to span produce large, flat-looking radii, which is why shallow segmental arches need such long trammel arms.
"Arc length is basically the same as the span." The arc is always longer than the chord it spans. Because the arc curves above the straight springline, its length exceeds the span — sometimes only slightly for shallow arches, but substantially for a full semicircle, so ordering trim by span alone will come up short.
Related calculators
Frequently asked questions
How do I calculate the radius of an arch from its span and rise?
Radius = (rise² + (span/2)²) ÷ (2 × rise). A 48-inch span with a 6-inch rise: (6² + 24²) ÷ (2×6) = (36+576) ÷ 12 = 51 inches.
What is the rise of an arch?
Rise is the height of the curve measured from a straight line drawn across the span (the springline) up to the highest point of the arc. A shallow segmental arch has a small rise relative to its span; a tall Roman-style arch has a rise close to half the span.
How much trim or track material do I need for an arch?
The arc length — the curved distance along the top of the opening — tells you how much flexible trim, track, or casing to buy. It's found from the radius and the angle the arc sweeps through, using arc length = radius × angle in radians.
How do I mark the arch curve for cutting?
Once you know the radius, you can strike the arc with a trammel (a beam compass) pivoted at a point directly below the center of the span, at a distance equal to the radius down from the springline — or minus the rise if the center falls above it, depending on the radius and rise relationship.
Sources & references
This tool uses standard formulas and reference values from:
- • American Concrete Institute — ACI 318, Building Code Requirements for Structural Concrete. concrete.org
- • ICC — International Residential Code (IRC), span, footing and framing tables. codes.iccsafe.org
- • APA – The Engineered Wood Association, allowable span and load guidance.
Estimates for planning only. Span, load and code values vary by jurisdiction — verify against your local adopted code and a licensed engineer before building.