Cutting a curved doorway or window opening without accurate measurements is one of the fastest ways to ruin good lumber. A slightly lopsided arch throws off trim work, header framing, and drywall returns for the rest of the project, and there's no easy way to fix an arch that's cut wrong after the fact.

The arch calculator solves this by turning your desired span and rise into the exact geometry you need to mark, cut, or frame a true elliptical arch. Whether you're a DIYer finishing a hallway opening or a finish carpenter laying out a custom window, this tool gives you the string length, foci placement, and curve points needed to get a smooth, symmetrical arch on the first try.

What Is an Arch Calculator and Why Does It Matter?

An arch calculator determines the exact curve geometry, span, rise, and radius points needed to lay out an accurate elliptical arch for a doorway, window, or decorative opening. It matters to homeowners planning a renovation, finish carpenters framing custom openings, and cabinet makers building arched doors and panels.

The core method uses ellipse geometry, treating the arch's width as the major axis and its height as the minor axis, then calculating focus points for the classic string-and-pins layout method. Get this wrong and the arch comes out lopsided, uneven on each side, or mismatched with the door or window unit it's meant to frame.

What You Need Before You Start

Gather these measurements before you start marking any lumber or drywall.

The full span of the opening, measured wall to wall at the springline where the curve begins.
The desired rise, meaning how tall you want the curve to reach above the springline at its peak.
Whether you're building a full elliptical arch or a segmental arch that only curves partway.
The thickness of the material you're cutting the template or arch form from.
Whether the arch needs to match an existing door, window, or millwork unit's exact curve.
Available workspace, since the string method needs enough flat surface to swing the full ellipse.

How to Use This Arch Calculator — Step by Step

Follow these steps in order to generate an accurate, ready-to-cut arch layout.

Measure the full span of your opening at the springline and enter it as your width.
Measure or decide your desired rise, the height of the curve above the springline.
Enter both numbers into the calculator to generate the semi-major and semi-minor axis lengths.
Note the calculated focus point distance, which tells you where to place your marking pins.
Cut a piece of string or twine to the calculated loop length for the trammel method.
Mark the two foci points on your material, then loop the string around both pins and a pencil.
Pull the string taut and trace the full curve, keeping consistent tension the entire way around.

The Arch Calculator Formula — Explained Simply

An elliptical arch is just half of a full ellipse, so the math starts with standard ellipse geometry using your span as the long axis and your rise as the short axis. Once you know those two axis lengths, you can calculate exactly where to place two pins so a taut string traces a perfect curve between them.

c = √(a² − b²), where a = Span ÷ 2 and b = Rise

Span — the full width of the opening at the springline, in inches, typically 24 to 96 for residential doorways and windows.
Rise — the height of the arch's peak above the springline, in inches, typically 6 to 36 depending on the desired curve depth.
a (semi-major axis) — half the span, in inches, defining how far the ellipse stretches side to side.
b (semi-minor axis) — the rise, in inches, defining how tall the ellipse stretches at its center.
c (focus distance) — the distance from the arch's center to each pin location, in inches, always smaller than a.

This works because an ellipse is defined by two fixed points, called foci, where the sum of distances from any point on the curve to both foci stays constant. That constant sum equals the string loop length, which is why looping a fixed length of string around two pins and a pencil traces a mathematically perfect ellipse every time. This is a widely accepted industry method rooted in classical ellipse geometry, long used in fine carpentry and timber framing under the name of the trammel or gardener's ellipse method.

Arch Calculator — Worked Examples

Beginner Scenario: Standard Interior Doorway Arch

A homeowner in Ohio wants to add a decorative elliptical arch to a 36 inch wide interior doorway opening, with a rise of 9 inches above the springline.

Calculate the semi-major axis: a = 36 ÷ 2 = 18 inches.
The semi-minor axis equals the rise directly: b = 9 inches.
Calculate the focus distance: c = √(18² − 9²) = √(324 − 81) = √243 ≈ 15.6 inches.
Calculate the string loop length, equal to 2a: 2 × 18 = 36 inches, plus a small allowance for tying knots.
Place two pins 15.6 inches apart from center, or 31.2 inches apart total, along the springline.

This gives the homeowner a clean, gentle arch that fits a standard doorway without looking overly dramatic. A 36 inch string loop with pins spaced 31.2 inches apart traces the curve in one smooth pass, ready to transfer onto a plywood template.

Professional Scenario: Custom Window Arch for a Finish Carpenter

A finish carpenter is building a custom arched window casing to match an existing 60 inch wide window opening with a specified rise of 15 inches, matching an architect's drawing.

Calculate the semi-major axis: a = 60 ÷ 2 = 30 inches.
The semi-minor axis equals the rise: b = 15 inches.
Calculate the focus distance: c = √(30² − 15²) = √(900 − 225) = √675 ≈ 26 inches.
Calculate the string loop length: 2 × 30 = 60 inches, plus roughly 4 inches of slack for tying.
Mark pins 26 inches apart from center, or 52 inches apart total, on the plywood template stock.
Trace the full curve with a pencil looped in the 64 inch string, keeping the string taut throughout.

The carpenter now has a template that will match the architect's specified curve exactly, since true ellipse geometry guarantees a smooth, symmetrical result rather than an approximated arc. This template becomes the master pattern for both the window casing and any matching interior trim on the same job.

Edge Case: A Very Flat Segmental-Elliptical Arch Over a Wide Opening

A contractor needs a very shallow, low-profile arch over an 84 inch wide great room opening, with a rise of only 6 inches, since the ceiling height leaves little room for a deep curve.

Calculate the semi-major axis: a = 84 ÷ 2 = 42 inches.
The semi-minor axis equals the rise: b = 6 inches.
Calculate the focus distance: c = √(42² − 6²) = √(1,764 − 36) = √1,728 ≈ 41.6 inches.
Calculate the string loop length: 2 × 42 = 84 inches.
Mark pins 41.6 inches apart from center, or 83.2 inches apart total, almost spanning the full width.

This case shows how a shallow rise relative to a wide span pushes the foci almost all the way out to the edges of the ellipse. The contractor needs a very long, flat working surface to swing this curve accurately, and should double check that the string stays taut near the ends, where a flat ellipse is most prone to small tracing errors.

Regional Variant: Metric Arch Layout for a Custom Millwork Shop

A millwork shop in Montreal, Quebec is building an arched cabinet door with a span of 90 centimeters and a rise of 20 centimeters, and needs all layout numbers in metric units for their shop drawings.

Calculate the semi-major axis: a = 90 ÷ 2 = 45 centimeters.
The semi-minor axis equals the rise: b = 20 centimeters.
Calculate the focus distance: c = √(45² − 20²) = √(2,025 − 400) = √1,625 ≈ 40.3 centimeters.
Calculate the string loop length: 2 × 45 = 90 centimeters, plus roughly 5 centimeters of slack for knots.
Mark pins 40.3 centimeters apart from center, or 80.6 centimeters apart total, on the shop's layout table.

Since the underlying ellipse math doesn't care what unit system you use, this shop can run the exact same formula in centimeters without any conversion headaches. The result is a precise curve for the cabinet door pattern, ready to trace directly onto the door blank stock.

Arch Calculator Results — What the Numbers Mean

A low rise relative to span, generally under 20 percent of the total width, produces a shallow, gentle curve often called a segmental-elliptical arch, common in transitional or craftsman-style homes. A higher rise, approaching 40 to 50 percent of the span, produces a dramatic, rounded curve more typical of Mediterranean or Georgian architectural styles.

The normal residential range for interior doorway arches runs between 15 and 30 percent of span for rise, giving a curve that reads as decorative without overwhelming standard 8-foot ceiling heights. Exterior entryways and formal window openings often push rise higher, since taller walls and grander proportions can support a deeper curve without looking cramped.

The single most common misunderstanding is assuming any curve drawn by hand or eye counts as a true ellipse. A freehand or compass-arc curve almost always looks slightly egg-shaped or lopsided once installed, while the string-and-pins method guarantees a mathematically true, symmetrical ellipse every time.

If your rise-to-span ratio comes out very low, under about 10 percent, double check your working surface is long enough, since the foci sit close to the very edges of the span and leave little room for pin placement error.

How Your Inputs Change the Result

Span has the most direct effect on your layout, since it sets the entire semi-major axis and scales every other measurement proportionally. Doubling your span from 36 to 72 inches at the same rise doesn't just double the string length, it also pushes the focus points much farther apart, changing how the curve feels visually even at an identical rise.

Rise changes the curve's character more than any other input, since a small change in rise at a fixed span can shift an arch from barely-there to dramatically rounded. A 36 inch span with a 6 inch rise looks almost flat, while the same 36 inch span with an 18 inch rise looks nearly semicircular.

Span | Rise | Focus Distance | Curve Character 36 in | 6 in | 17.5 in | Very shallow, subtle curve 36 in | 12 in | 15.0 in | Moderate, classic doorway arch 36 in | 18 in | 0 in | Perfect semicircle 36 in | 24 in | Not a valid ellipse | Rise cannot exceed half the span

The focus distance shrinks as rise grows relative to span, and it hits zero exactly when rise equals half the span, since that's the mathematical point where the ellipse becomes a perfect circle. Pushing rise any higher than half the span makes an ellipse impossible, meaning you'd need a different arch type entirely, like a pointed Gothic or a stilted arch.

Material thickness doesn't change the ellipse geometry itself, but it does affect how you transfer the curve. Thicker template stock needs a slightly longer pencil offset from the string loop to keep the trace accurate, which matters most on thick plywood or MDF template blanks.

Arch Calculator — Common Mistakes to Avoid

Confusing rise with total arch height, when rise specifically means the curve's height above the springline, not the full opening height including the straight sides below it.
Assuming a compass can draw a true ellipse, when a compass only produces circular arcs, which look visibly different from a proper elliptical curve once installed.
Cutting the string loop to exactly 2a without adding slack for knots, resulting in a loop that's too tight to trace the full curve accurately.
Placing pins at the wrong distance by using the full focus-to-focus distance instead of the correct center-to-focus distance for each pin.
Trying to build an arch where the rise exceeds half the span, which isn't mathematically possible as a true ellipse and needs a different arch style.
Ignoring wood grain direction when cutting the arch template, which can cause the curved trim piece to be structurally weak at its thinnest points.
Working on a surface too short for the full ellipse, causing pins to fall off the edge of the material during layout.
Mixing imperial and metric measurements mid-project, which throws off both the string length and the pin spacing calculations.

Key Terms Explained

Span

The full width of the arch opening measured at the springline, where the straight sides meet the curve.

Rise

The height of the arch's curve measured from the springline up to the highest point of the arc.

Springline

The horizontal line where the straight vertical sides of an opening end and the curved arch begins.

Semi-Major Axis

Half the longer axis of an ellipse, corresponding to half the span in an arch layout.

Semi-Minor Axis

Half the shorter axis of an ellipse, corresponding directly to the rise in an arch layout.

Foci

The two fixed points inside an ellipse used in the trammel method, where the string loop anchors during tracing.

Trammel Method

A classical layout technique using a string loop around two foci pins and a pencil to trace a true ellipse.

Segmental Arch

An arch using only a shallow portion of a curve, rather than a full semicircular or semi-elliptical sweep.

How This Calculator Works — Methodology

This calculator applies standard ellipse geometry, calculating the semi-major axis from half the span and the semi-minor axis directly from the rise, then deriving focus distance using the classical relationship c equals the square root of a squared minus b squared. It assumes a true half-ellipse arch shape, which is the most common form used in residential doorway and window openings.

The calculation excludes Gothic, pointed, and stilted arch types, since those follow entirely different geometric construction methods rather than ellipse math. It also assumes the springline is perfectly level and the opening is symmetrical, since an out-of-level or irregular opening needs adjusted, site-specific measurements before applying this formula. Results are typically presented to one decimal place, since carpentry layout work rarely benefits from more precision than a standard tape measure or ruler can reliably mark.

Frequently Asked Questions About Arch Calculator

1. What does an arch calculator actually calculate?

It calculates the exact ellipse geometry, including focus point placement and string loop length, needed to draw a true elliptical arch from your desired span and rise. This gives you a ready-to-use layout for cutting templates or marking openings.

2. How accurate is an arch calculator for real construction layouts?

It's mathematically exact, since it's based on true ellipse geometry rather than an approximation. Real-world accuracy depends on how carefully you mark pins and maintain string tension while tracing the curve.

3. What is the arch layout formula in simple terms?

Take half your span and your full rise, then use those two numbers to calculate how far apart to place two pins using the formula c equals the square root of a squared minus b squared. A string looped around those pins traces the exact curve.

4. Does the arch formula work the same way in metric measurements?

Yes, the underlying ellipse geometry is unit-independent, so you can run the exact same formula in centimeters or millimeters without any conversion factor. Just keep your span and rise in the same unit system throughout the calculation.

5. When does this calculator not apply?

It doesn't apply to Gothic, pointed, or stilted arches, since those shapes follow entirely different geometric construction rules than a true ellipse. It also doesn't apply if your rise exceeds half your span, since that isn't a valid ellipse shape.

6. How do I get a more precise arch layout?

Use a longer, more rigid string like waxed nylon cord instead of stretchy twine, since stretch introduces small tracing errors near the ends of a shallow arch. Also work on the largest flat surface available to avoid pins slipping off the edge.

7. What's a typical rise-to-span ratio for a residential arch?

Most interior doorway arches use a rise between 15 and 30 percent of the total span, giving a decorative curve that fits comfortably under standard 8-foot ceilings. Exterior and formal openings often run higher, closer to 30 to 40 percent.

8. Should I consult a professional before cutting a structural arched opening?

For decorative trim arches, careful DIY layout using this calculator is usually sufficient. For structural openings that remove load-bearing framing, always consult a licensed contractor or structural engineer before cutting.

Related Calculators Worth Bookmarking

Rafter Length Calculator — useful when your arched opening sits below a roofline that also needs precise angle cuts.
Stair Stringer Calculator — helps with adjacent layout work when an arched opening leads to a staircase.
Board Foot Calculator — estimates how much lumber you need for the template and trim pieces surrounding your arch.
Circle Calculator — a simpler alternative when you actually want a true semicircular arch instead of an ellipse.
Miter Angle Calculator — helps cut the straight trim pieces that meet your curved arch at the springline.
Window Trim Calculator — useful for estimating casing material once your arched window layout is finalized.
Plywood Sheet Calculator — helps figure out how many sheets you need for cutting a full-size arch template.
Roof Pitch Calculator — relevant when an arched opening sits on an exterior wall with a specific roof slope above it.

References and Further Reading

Fine Homebuilding. Laying Out an Elliptical Arch. Taunton Press, 2019.
American Institute of Timber Construction. Timber Construction Manual. Wiley, 2012.
National Association of Home Builders. Residential Construction Performance Guidelines. NAHB, 2020.
The Journal of Light Construction. Framing Curved Openings. Zweig Group, 2021.

Important Disclaimer

This calculator is for informational and educational purposes only and provides a geometric layout estimate, not a structural engineering assessment. This tool does not constitute professional construction or safety advice. Always consult a qualified contractor or structural engineer before cutting load-bearing framing or structural openings.