Arch Calculator
ConstructionCalculate the radius and arc length of a segmental arch from its span and rise. Get precise measurements for framing, masonry, and trim carpentry projects.
Radius
30
Arc Length
55.64
What is a Arch?
An Arch Calculator determines the radius and arc length of a segmental (circular) arch based on its span and rise — the two dimensions most easily measured on an actual doorway, window, or porch opening. Segmental arches appear throughout residential and commercial construction, from decorative window headers to structural masonry openings, and getting the curve geometry right is essential before cutting templates, ordering trim, or laying brick voussoirs. Rather than relying on trial-and-error curve tracing, this calculator applies standard circular segment formulas to give you exact numbers.
The radius tells you where to anchor a trammel or compass to draw the arc precisely, while the arc length tells you how much curved material — trim, flexible molding, or masonry coursing — you'll actually need, which is always more than the straight-line span. If you're planning the broader framing for an arched opening, the Framing Calculator and Board Foot Calculator help estimate the surrounding lumber needs for the same project.
How to use this Arch calculator
- Measure the Span of your arch opening — the straight horizontal distance across the base — and enter it in inches.
- Measure or specify the Rise — the vertical height of the arc's curve above the baseline — and enter it in inches.
- Read the Radius result, which tells you the distance to set on a trammel or compass to trace the arc accurately.
- Read the Arc Length result to determine how much curved trim, molding, or masonry material you'll need to order.
- Use the radius to locate your pivot/center point on your workpiece before tracing or routing the curve.
- Adjust the span or rise values to compare how different arch proportions change the radius and material requirements before committing to a final layout.
Formula & Methodology
The calculator uses standard circular segment geometry to solve for the radius and arc length of a segmental arch: Radius = (Span² ÷ (8 × Rise)) + (Rise ÷ 2) Arc Length = 2 × Radius × asin((Span ÷ 2) ÷ Radius) Where asin is the inverse sine function (calculated in radians), Span and Rise are both measured in inches, and the resulting Arc Length is also in inches. Worked example: A doorway with a 48-inch span and a 12-inch rise. Radius = (48² ÷ (8 × 12)) + (12 ÷ 2) = (2,304 ÷ 96) + 6 = 24 + 6 = 30 inches. Arc Length = 2 × 30 × asin(24 ÷ 30) = 2 × 30 × asin(0.8) = 2 × 30 × 0.9273 radians ≈ 55.64 inches — meaning you'd need to order roughly 4.6 linear feet of curved trim for this arch, not just the 4-foot span.
Frequently Asked Questions
A segmental arch is a curved architectural element formed by a single arc that is less than a full semicircle, defined by its span (the horizontal opening width) and its rise (the vertical height of the curve above the springline). Segmental arches are common over doorways, windows, and porch openings because they are structurally efficient and easier to build than full semicircular arches. This calculator determines the radius and arc length needed to lay out that curve accurately.
The calculator takes the span and rise you enter and uses circular geometry to solve for the radius of the arc that would produce that exact span and rise. Once the radius is known, it calculates the arc length — the actual curved distance along the arch — using the inverse sine function. Both values are essential for cutting arch templates, ordering curved trim, or laying out masonry voussoirs.
Radius equals (span squared divided by 8 times the rise) plus half the rise: R = (S² ÷ 8R) + (R ÷ 2). Arc length equals 2 times the radius times the arcsine of half the span divided by the radius: L = 2R × asin(S ÷ 2R). These formulas come from standard circular segment geometry used throughout carpentry and masonry trades.
Span is the straight-line horizontal distance across the bottom of the arch opening, while rise is the vertical height from that baseline to the highest point of the curve. Radius is a derived value — the distance from the arc's curve to its theoretical center point — and is not something you measure directly on a finished arch. This calculator solves for radius and arc length from the span and rise you can easily measure on site.
Arc length tells you the actual curved distance along the arch, which is different from the straight-line span and is critical for ordering the correct amount of curved trim, flexible molding, or masonry units. If you only used the span to estimate materials, you would consistently underorder because the curve is always longer than the straight-line distance it spans. Getting the arc length right avoids costly reorders and mismatched trim pieces mid-project.
Enter your arch opening's Span, which is the horizontal width at the base of the arch, and the Rise, which is how high the curve rises above that baseline. The calculator instantly returns the Radius of the arc and the total Arc Length. Use the radius value to set your trammel or router jig, and use the arc length to calculate material needs for trim or masonry.
Yes, a semicircular arch is simply a special case where the rise equals exactly half the span, and the formula works correctly for that case as well. For a true semicircle, the calculated radius will equal half the span. This calculator handles both shallow segmental arches and full semicircular arches using the same geometry.
Most carpenters use a trammel — a beam compass with a pivot point and pencil set at the calculated radius distance — anchored at the arc's center point to trace the curve directly onto plywood or drywall. Alternatively, a string-and-pencil method works for larger arches where a rigid trammel beam isn't practical. Either method requires knowing the precise radius this calculator provides to locate the pivot point correctly.
Yes, a lower rise relative to the span produces a flatter, more elongated arch with a larger radius, while a higher rise produces a rounder, more pronounced curve with a smaller radius. Architectural style often dictates this ratio — Roman and Romanesque arches tend toward higher rises, while many contemporary designs use shallow, flat segmental arches. Adjusting the rise value in this calculator lets you compare how different proportions affect the resulting radius and arc length.
This calculator works in inches for both span and rise, which is the standard unit carpenters and finish trades use for door, window, and trim layout work. The radius and arc length outputs are also given in inches to match. If you're working from a plan in feet, convert to inches first by multiplying by 12 before entering your values.
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