How to use the arch template calculator
- Choose a circular arc or a half ellipse. A circular arc has one radius, while a half ellipse is flatter at the top.
- Enter the span, the width of the opening, and the rise, the height at the center above the span line.
- Enter the equal spaces to divide the span into. Eight spaces give nine marks.
- Mark each distance along the span line, mark the height straight up from it, and bend a thin strip through the points to draw the curve.
Formula
circular arc: radius = span^2 / (8 x rise) + rise / 2
height at distance x from the left end = sqrt(radius^2 - (x - span / 2)^2) - (radius - rise)
curve length = 2 x radius x arcsin(span / (2 x radius))
half ellipse: height = rise x sqrt(1 - ((x - span / 2) / (span / 2))^2)
The height of a circular arc comes from placing the circle center below the span line by radius minus rise. The length of an ellipse has no simple formula, so it is found by numerical integration.
A 36 inch opening with a 9 inch rise
The radius is 36^2 / (8 x 9) + 9 / 2 = 18 + 4.5 = 22.5 inches. Marks every 4.5 inches have heights 4.5 at the first mark, then 7.122, 8.545 and 9 at the center. The arc spans an angle of 106.26 degrees at its center and is 41.73 inches long.
As a half ellipse with the same span and rise, the heights at the same marks are 5.953, 7.794, 8.714 and 9, and the curve is 43.60 inches long. The ellipse rises faster near the ends and is flatter across the top.
Drawing the curve
Lay a straight board or string along the span line, mark the distances, and measure each height up from the line with a square. Hold a thin flexible strip against the marks and trace along it. More marks give a smoother curve, and 8 or more is usually enough for an opening under a few feet.
If you know three points on a circle instead of the span and rise, use the Circle From Three Points Calculator. For the area or circumference of the full circle, use the Circle Calculator.
Limits of a circular arch
A circular arch cannot rise higher than half its span, because a taller arch would bulge wider than its own opening. The tool asks you to lower the rise or choose the half ellipse. The curve length and radius apply to the center line of the curve, so add thickness and joint allowances for the real pieces.
More layout tools
For other cuts, curves and fitting, try the Cut List Optimizer, Plank Stagger Planner and Tile Layout Calculator.
Assumptions and limits
- The arch is symmetric, with the highest point at the middle of the span.
- A circular arch has a rise no greater than half the span.
- Heights are measured at right angles to the span line.
- Material thickness, joints and framing are not included.
Frequently asked questions
How do I find the radius of an arch from its span and rise?
Square the span, divide by eight times the rise, then add half the rise. A 36 inch span with a 9 inch rise has a radius of 22.5 inches.
What are ordinates in an arch template?
They are the heights of the curve above the span line at evenly spaced distances along it. Marking them and joining the points draws the arch.
What is the difference between a circular and an elliptical arch?
A circular arch has one radius everywhere. An elliptical arch has a tighter curve near the ends and is flatter on top, so for the same span and rise it is slightly longer.
How many marks should I use?
Eight spaces are fine for a small opening. Use more for a large arch or when you need a very smooth line, since a strip held through more points follows the curve better.