How do you find the radius of an arch?
Measure the width and how far it bows, and the radius follows — a 36" opening rising 4" needs a 42 1/2" radius. The sagitta formula, and a table of common arches.
A 36" opening bowing 4" at the middle is struck from a radius of 42 1/2".
The two measurements you can actually take — the width across and the rise in the middle — are all it needs. The centre point is 38 1/2" below the top of the arc, which is usually somewhere below the floor.
How that is worked out
- Call the half-width h and the rise r. The radius, the half-chord and (radius − rise) form a right triangle, so R² = h² + (R − r)².
- Solved for R that is (h² + r²) ÷ 2r: (18² + 4²) ÷ (2 × 4) = 42 1/2".
- This is the sagitta formula — sagitta being the Latin for arrow, because the rise is the arrow and the chord is the bowstring. It is the practical way to lay out an arch, because nobody can measure to a centre point that is six feet underground.
Radius for common openings, by how far the arch rises
| Opening width | 2" rise | 4" rise | 6" rise | 9" rise |
|---|---|---|---|---|
| 24" | 37" | 20" | 15" | 12 1/2" |
| 30" | 57 1/4" | 30 1/8" | 21 3/4" | 17" |
| 36" | 82" | 42 1/2" | 30" | 22 1/2" |
| 42" | 111 1/4" | 57 1/8" | 39 3/4" | 29" |
| 48" | 145" | 74" | 51" | 36 1/2" |
| 60" | 226" | 114 1/2" | 78" | 54 1/2" |
| 72" | 325" | 164" | 111" | 76 1/2" |
Worth knowing
- Striking the arc needs a trammel — a stick pivoting on a nail at the centre — and the centre is usually off the workpiece and often below the floor. A long straight batten clamped to a temporary frame is the usual workaround.
- For a shallow arch over a wide opening the radius gets very large very quickly: a 72" opening rising 2" needs a radius over twenty-seven feet, which is a batten nobody has. Bending a thin lath through three nails is the practical method there.
- A half-round arch is the special case where the rise equals half the width, and the radius is then just half the width. Everything shallower than that is what this table is for.
- Cut the arc in the workpiece, then use that piece as the template for every other one. Two arcs struck separately will not match, however careful the arithmetic.
Work out your own numbers
- Circle and arc layout
A trammel arm's length for a full circle, or the radius of a shallow arc from its chord and rise.
- Grid enlargement calculator
Scale a small pattern up to full size using the classic square-by-square grid method.
- Miter angle calculator
The miter for baseboard, casing or chair rail at any corner angle — flat trim, so there's no bevel to set.
Related questions
- How do you calculate the radius of an arc?
- Take half the width and call it h, take the rise at the middle and call it r, then the radius is (h² + r²) ÷ 2r. It is the sagitta formula, and it works from the two measurements you can actually take on a real opening — the width across and how far the arc bows up.
- How do I draw a large arc without a compass?
- Use a trammel — a batten with a nail through one end as a pivot and a pencil at the radius. When the radius is too large for that, bend a thin lath through three points (the two ends of the chord and the top of the rise) and trace along it. The lath naturally takes a curve close enough to a true arc for trim work.
- What rise should an arch over a door have?
- There is no rule — it is proportion. A shallow 2" rise over a 36" opening reads as a subtle eyebrow; 6" reads as a deliberate arch; half the width is a full semicircle. Draw it full size on cardboard and stand back before committing, because the difference between good and odd is a couple of inches.