Arch Calculator
Lay out an arched doorway, window, pass-through or garden arbor. Enter the span and rise, pick Segmental, Half-round or Gothic, and get the radius, where to set your trammel pivot, the arc length, opening and soffit area, outer trim radius and stock length, plus layout points you can mark straight onto the framing — with a by-rise comparison, a profile chart and a downloadable PDF.
| Point | From left jamb | Height above springing | Decimal |
|---|---|---|---|
| Enter an arch to see layout points | |||
| Rise | Radius | Arc length | Opening area |
|---|---|---|---|
| Enter an arch to compare | |||
How to use the arch calculator
Four quick steps turn an opening width and rise into the radius, pivot point, arc length, trim length and layout points you need to draw and build the arch — with by-rise tables and a downloadable report.
Pick the arch type
Choose a segmental (shallow curve), half-round or Gothic (pointed) arch.
Enter span & rise
Give the opening width between jambs and how high the curve rises above the springing line.
Add depth & trim
Enter the wall or jamb depth for the soffit and the casing width for trim length, then pick 6, 8 or 12 layout stations.
Read results & save a PDF
Get the radius, pivot, arc, areas, trim stock and layout heights — then download a report.
How the arch radius is calculated
An arch is a slice of a circle, so its span (the chord) and rise fix the radius. Everything else — pivot, arc, area and trim — follows from that radius. Here is the exact math the calculator runs.
Worked example — 36″ segmental opening, 9″ rise, 4 1/2″ wall, 3 1/2″ casing
Radius: (36² ÷ 4 + 9²) ÷ (2 × 9) = 405 ÷ 18 = 22 1/2" · Pivot: 22 1/2 − 9 = 13 1/2" below the springing line
Angle: 2 × acos(13.5 ÷ 22.5) ≈ 106.3° · Arc: 22.5 × 1.855 ≈ 41 3/4" · Area: ≈ 1.57 sq ft
Soffit: 41.73 × 4.5 ≈ 1.30 sq ft · Trim: outer radius 26", outer arc 53 5/16" × 1.10 ≈ 58 5/8"
≈ 22 1/2″ radius swung from 13 1/2″ below the springing line — a 41 3/4″ curve that needs about 4′ 10 5/8″ of casing.
Arch radius reference charts
Handy lookups for how rise changes a segmental arch, how Gothic arches grow taller, and the layout heights for the worked example.
| Rise | Radius | Arc length | Opening area |
|---|---|---|---|
| 3″ (1/12) | 55 1/2″ | 36 11/16″ | 0.50 sq ft |
| 4 1/2″ (1/8) | 38 1/4″ | 37 1/2″ | 0.76 sq ft |
| 6″ (1/6) | 30″ | 38 5/8″ | 1.02 sq ft |
| 9″ (1/4) | 22 1/2″ | 41 3/4″ | 1.57 sq ft |
| 12″ (1/3) | 19 1/2″ | 45 7/8″ | 2.17 sq ft |
| 18″ (half-round) | 18″ | 56 9/16″ | 3.53 sq ft |
A rise of about 1/4 to 1/6 of the span looks balanced. Below 1/12 the arch reads as nearly flat.
| Rise | Radius | Arc length | Opening area |
|---|---|---|---|
| 18″ (0.5 × span) | 18″ | 56 9/16″ | 3.53 sq ft |
| 21 1/2″ (0.6) | 21 13/16″ | 60 7/8″ | 4.04 sq ft |
| 27″ (0.75) | 29 1/4″ | 68 13/16″ | 4.88 sq ft |
| 31 1/4″ (equilateral) | 36 1/8″ | 75 1/2″ | 5.54 sq ft |
| 36″ (1.0) | 45″ | 83 7/16″ | 6.29 sq ft |
Radius equal to the span is an equilateral arch; larger is a lancet, smaller is a drop arch. A Gothic rise must be at least half the span.
| From left jamb | Height above springing line |
|---|---|
| 0″ / 36″ | 0″ |
| 4 1/2″ / 31 1/2″ | 4 1/2″ |
| 9″ / 27″ | 7 1/8″ |
| 13 1/2″ / 22 1/2″ | 8 9/16″ |
| 18″ (crown) | 9″ |
Mark each height, then bend a thin batten through the points and trace the curve.
What the arch calculator gives you
A span and a rise give you everything you need to draw, frame and trim the arch.
Arch layout at a glance
Who uses an arch calculator
From a drywall archway to a brick garden entry, the same radius math lays out the curve.
Turning a square opening into a drywall arch and needs the radius, pivot and how much bendable drywall to buy.
- Enter span and rise
- Swing the arc from the pivot
- Use the soffit area
Making arched casing or a jamb and wants the outer radius, arc length and stock length with waste.
- Enter casing width
- Read outer radius
- Order trim stock
Building a centering form for a brick, stone or timber arch and wants exact layout points to cut the form.
- Pick 12 stations
- Cut a plywood template
- Size abutments with a pro
Tips for laying out a clean arch
Good arches come from careful layout — these habits keep the curve true and both sides matched.
Arch calculator FAQ
Quick answers about arch radius, layout, rise, trim and when an arch needs structural support.
For a segmental arch, the radius is (span² ÷ 4 + rise²) ÷ (2 × rise). A 36″ opening with a 9″ rise gives (324 + 81) ÷ 18 = 22 1/2″.
A half-round arch is simply half the span, and a Gothic arch uses (span² ÷ 4 + rise²) ÷ span. The calculator does this for you and rounds to the nearest 1/16″.
For a segmental arch, the pivot sits on the centerline, radius minus rise below the springing line — 13 1/2″ below it for a 36″ span with a 9″ rise. Screw a temporary block there and swing a trammel set to the radius.
A half-round arch pivots right on the springing line at the center. A Gothic arch uses two pivots on the springing line, each one radius in from the opposite jamb.
A segmental arch is a shallow slice of a circle with a rise less than half the span. A half-round arch is a full semicircle whose rise equals half the span.
A Gothic arch is two arcs meeting at a point, so its rise is at least half the span. When the radius equals the span it is an equilateral arch, a larger radius makes a lancet and a smaller one a drop arch.
A rise of about 1/4 to 1/6 of the span usually looks balanced. For a 36″ doorway that is roughly a 6″ to 9″ rise.
Below about 1/12 of the span the arch looks almost flat, and in masonry it needs strong abutments or a steel lintel.
Use the layout points instead. The calculator gives the arch height at 6, 8 or 12 equal stations across the span — for a 36″ span with a 9″ rise and 8 stations they are 0, 4 1/2″, 7 1/8″, 8 9/16″ and 9″ at the crown, mirrored on the other side.
Mark each height, bend a thin batten through the points and trace the curve. This is handy for wide, shallow arches — a 72″ span with a 12″ rise has a 5 ft radius.
Trim follows the outer edge, so its radius is the arch radius plus the casing width. With a 22 1/2″ radius and 3 1/2″ casing the outer radius is 26″ and the outer arc is about 53 5/16″.
The calculator adds 10% for waste, so plan on about 58 5/8″ (4′ 10 5/8″) of trim stock for the curve.
Multiply the arc length by the wall or jamb depth. A 41 3/4″ arc in a 4 1/2″ deep wall is about 1.30 sq ft.
Use bendable (1/4″) drywall or thin plywood for the curved soffit, and add corner bead or a flexible trim along the edges.
This calculator is a layout aid only. Cutting an arch into a bearing wall still needs a properly sized header above it, and the arch itself is usually non-structural framing or drywall.
Masonry arches carry load through their shape and need adequate abutments, so have them designed by a qualified professional.
Layout aid only — not a structural design. This calculator works in inches and handles arch geometry only. Openings in bearing walls need a properly sized header, and masonry or load-bearing arches should be designed by a qualified professional. Always cut a plywood or hardboard template and test-fit it before cutting framing, trim or stone.