Arch Calculator
The radius, arc length and centre point of a segmental arch from its span and rise, and the height of the curve at any point for laying it out.
Result
30 in
Set your trammel to thisGeometry is exact; lengths are read to the nearest sixteenth of an inch.
- Arc length
- 38 5/8 in
- Centre point below the springing line
- 24 in
- Angle the arc turns through
- 73.74 °
- Area under the arc
- 147.15 in² (1.02 ft²)
- Height of the curve at the layout point
- 4 5/8 in
https://calcgallery.com/lumber-carpentry/arch-calculator/
Results are estimates for planning only. Actual needs vary with site conditions, products and workmanship. Always confirm quantities with your supplier and follow local building codes and manufacturer instructions. Read the full disclaimer
How to use the Arch Calculator
- Measure the span, the straight width of the opening between the two points where the curve starts.
- Decide the rise, the height of the crown above that line, up to half the span.
- Read the radius to set a trammel or string to, and where its centre sits below the springing line.
- Optionally enter a distance from the centre line to get the height of the curve there, for marking points by hand.
How it is calculated
radius = (span² ÷ 4 + rise²) ÷ (2 × rise)angle = 4 × atan(2 × rise ÷ span)arc length = radius × angle (angle in radians)centre drop = radius − risearea = radius² × (angle − sin angle) ÷ 2height at x = √(radius² − x²) − centre dropThese are the circular segment relations: the span is the chord and the rise its height. The angle formula comes from the fact that the rise divided by half the span is the tangent of a quarter of the angle, which works for any segment up to a half circle.
Laying out by points
When a trammel is not practical, mark the curve by points: enter several distances from the centre line, one at a time, and mark the height of the curve at each. Join the marks with a flexible batten held against nails. Wider openings, masonry arches and anything carrying load need their own design; this page gives the geometry only.
With your numbers
- Radius = (36 in² ÷ 4 + 6 in²) ÷ (2 × 6 in) = 30 in
- Arc = radius × angle = 38.61 in
Quick reference
Geometry is exact; lengths are read to the nearest sixteenth of an inch.
| Job | Radius |
|---|---|
| 36 in span, 6 in rise | 30 in |
| 48 in half round | 24 in |
| 72 in span, 9 in rise | 76 1/2 in |
| 30 in span, 4 in rise | 30 1/8 in |
| 90 cm span, 15 cm rise | 75 cm |
| 120 cm half round | 60 cm |
Common questions about the Arch Calculator
- How do I find the radius of an arch from its width and height?
- Square half the span, add the rise squared, and divide by twice the rise. A 36 inch opening with a 6 inch rise has a radius of (18² + 6²) ÷ 12 = 30 inches, and the centre sits 24 inches below the springing line.
- How do I draw the arch on a board?
- Mark the centre line and measure down from the springing line by the centre distance shown. Fix a nail or pivot there and swing a trammel, a thin batten or a non-stretch string set to the radius, through both springing points and the crown. If the centre falls off the board, screw a scrap to it to hold the pivot.
- What if the rise is more than half the span?
- Then the curve is more than a half circle, a horseshoe arch, and its centre sits above the springing line. This page covers segmental and half-round arches, so it refuses that case.
- How long a strip do I need to follow the curve?
- The arc length is the distance along the curve from one springing point to the other. Add a little at each end to trim, and remember that a moulding or a bent lamination is measured along its own centre line, which lies on a slightly different radius.