Carpentry Calculators

Ellipse and elliptical arch calculator

From the width and the rise, where the two nails go for the string-and-pencil method, how long the string is, and the height of the curve at even stations across the opening so it can be plotted with a tape instead.

Shape
in

Across the springing line

in

Springing line up to the crown

Equal spaces for the offset table — more of them on a tight curve

Nails at the foci

20 13/16"each side of centre

String, nail to nail

48"

As a closed loop

89 9/16"

Length of the arch

58 1/8"

Name this job

Saved jobs stay in this browser. Nothing is sent anywhere, and a save is not a backup — a link is how you move one to another device.

F1F248" wide, foci 20 13/16" each side of centre

Drawing it with a string

NailsOn the springing line, one each side — 3 3/16" in from each end
20 13/16" from centre
String tied nail to nailBetween the knots, not counting what goes into them
48"
Or a closed loopAround both nails and the pencil
89 9/16"
Check the nailsFrom the crown to either nail should measure exactly this
24"
Area of the arch
3.14 sq ft

Use string that does not stretch — mason's line, not cotton — and keep the pencil upright. A string that gives a quarter inch under tension draws an arch a quarter inch too wide.

Or plot it with a tape

Mark each station along the springing line from the left end, square up the height at each one, and bend a thin batten through the points.

Height of the curve at each station
StationFrom left endHeight
00"0"
14"6 5/8"
28"8 15/16"
312"10 3/8"
416"11 5/16"
520"11 13/16"
624"12"
728"11 13/16"
832"11 5/16"
936"10 3/8"
1040"8 15/16"
1144"6 5/8"
1248"0"

About laying out an ellipse

Where do the nails go to draw an elliptical arch with a string?
On the springing line, one each side of centre at a distance c = √(a² − b²), where a is half the width and b is the rise. For a 48" opening with a 12" rise that is 20 13/16" each side of centre, or 3 3/16" in from each end. Those two points are the foci, and they are the only places the nails can go for a string to trace an ellipse.
How long should the string be?
Tied between the two nails, exactly the full width of the ellipse — 48" for a 48" arch — measured between the knots. Every point on an ellipse has the same total distance to the two foci, and that total is the long axis, which is why the pencil cannot leave the curve while the string is taut. Tied instead as a closed loop around both nails and the pencil, it is that plus the distance between the nails: 89 9/16". Use a line that does not stretch; any give draws the arch that much too wide.
How do I find the foci without doing the square root?
Set a tape or a stick to half the width — 24" for a 48" arch — hold one end at the crown, and swing the other down to the springing line. Where it lands on each side is a focus. It works because the distance from the top of the minor axis to either focus is always exactly half the long axis, and it doubles as a check on nails already driven.
Why use an ellipse rather than a circular arc?
Because of what happens at the ends. A circular arc with a 12" rise over 48" is part of a 30" radius circle, and it meets the springing line at an angle, leaving a visible kink where the arch meets the jamb. An ellipse of the same width and rise starts straight up off the springing line and flattens across the top, so it runs smoothly into vertical jambs. A shallow arch is where the difference shows most.
How long is the curve, for trim or a bending strip?
There is no exact formula for the perimeter of an ellipse, so the calculator uses Ramanujan's approximation, which is within a few parts per million — far closer than any tape. For a 48" arch with a 12" rise the curve is 58 1/8" long. That is the centreline of the curve, so add length for the miters or copes at each end and a few inches to hold on to while it is bent.

You might also need