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Arch Calculator

Construction

Calculate the radius and arc length of a segmental arch from its span and rise. Get precise measurements for framing, masonry, and trim carpentry projects.

Reviewed by the thecalcu.com team · Last updated July 10, 2026

6300
1100

Radius

30
Arc Length
55.64

This calculator computes your Radius, Arc Length from the values you enter.

Inputs
SpanRise
Outputs
RadiusArc Length

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.

Why Use an Arch Calculator?

Laying out an arch by eye or trial-and-error wastes material and time, especially when working with expensive trim stock or masonry units that can't easily be re-cut. This calculator solves that by giving you the exact radius needed to set a trammel or router jig before you make a single cut, and the arc length needed to order the right amount of curved material from the start.

For example, if you're building a segmental arch over a 48-inch-wide doorway with a 12-inch rise, the calculator instantly tells you the radius to set your trammel at and the total arc length to order in trim or brick. That precision matters most on masonry projects, where mis-cut voussoirs are essentially unusable, and on custom millwork, where flexible molding is sold and priced by the linear foot of curve.

Who Should Use This Calculator?

Finish carpenters building arched door and window casings need the exact radius to set up a router jig or trammel before cutting expensive trim stock. Masons laying segmental brick or stone arches need both radius and arc length to plan voussoir counts and coursing before starting work. DIY homeowners framing a decorative arch opening in drywall or trim can use this calculator to avoid guesswork when tracing the curve template.

General contractors estimating material costs for arched openings across multiple units in a development benefit from quick, repeatable radius and arc-length figures. Architectural millwork shops fabricating custom arched casings or transoms use these calculations daily to program CNC equipment or set manual layout tools. Pair this with the Framing Calculator when planning the structural rough opening around the finished arch.

What Insights Does the Arch Calculator Give You?

Radius is the primary output and tells you the distance from the arc's theoretical center point to the curve itself, this is the number you use to set a trammel, compass, or router jig radius when physically tracing or cutting the arch. Arc Length gives you the actual curved distance along the top of the arch, which is always longer than the straight-line span and is the figure to use when ordering curved trim, flexible molding, or calculating masonry unit counts.

Together, these two numbers turn a simple span-and-rise measurement into everything you need to physically produce the arch, the radius for layout, and the arc length for materials. A small rise relative to a wide span produces a large radius and a relatively flat arc, while a tall rise produces a small radius and a more pronounced curve, both of which this calculator handles with the same formula.

How to use this Arch calculator

  1. Measure the Span of your arch opening, the straight horizontal distance across the base, and enter it in inches.
  2. Measure or specify the Rise, the vertical height of the arc's curve above the baseline, and enter it in inches.
  3. Read the Radius result, which tells you the distance to set on a trammel or compass to trace the arc accurately.
  4. Read the Arc Length result to determine how much curved trim, molding, or masonry material you'll need to order.
  5. Use the radius to locate your pivot/center point on your workpiece before tracing or routing the curve.
  6. Adjust the span or rise values to compare how different arch proportions change the radius and material requirements before committing to a final layout.
Show formula & methodology ↓Show less ↑

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

What is a segmental arch in construction?
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.
How does the Arch Calculator work?
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.
What is the formula for arch radius and arc length?
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.
What is the difference between span, rise, and radius?
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.
Why do I need to know the arc length of an arch?
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.
How do I use the Arch Calculator?
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.
Can this calculator be used for a full semicircular arch?
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.
What tools do I need to physically draw the arch once I have the radius?
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.
Does the rise-to-span ratio affect how flat or round the arch looks?
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.
What units does the Arch Calculator use?
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.
Also known as
segmental arch calculatorarch radius calculatorarc length calculatorarch span and rise calculatorcircular arch calculatorarch layout calculator