How do I find the radius of an arch from its width and height?

Radius = (half the span squared + the rise squared) ÷ (2 × the rise). A 48-inch opening with an 8-inch rise: (24² + 8²) ÷ 16 = 640 ÷ 16 = 40 inches. Swing a 40-inch string or trammel from a point 32 inches below the springing line (40 − 8) and it draws the arch.

How do I lay out an arch without a string?

Snap the chord (the straight line across the bottom), mark stations along it from the centre, and measure up from each station the offset the calculator gives. Join the points with a flexible batten. It is how you lay out an arch on a sheet of plywood, a curved form, or a flower-bed edge too big for a string.

How long is the curved edge?

Arc length = radius × the angle in radians. The calculator gives it, which is what you need for the trim, the flexible track or the edging along the curve. For the 48 × 8 in arch it is about 51.5 inches.

What if the rise is more than half the span?

Then the curve is more than a half circle — the arch bulges wider than the opening. The formula still works and the offsets are still measured up from the chord.

Arc & radius calculator

Claro (cuerda)

La línea recta entre los extremos de la curva. Pulgadas, o pies y pulgadas como 4' 2".

Flecha (altura en el centro)

Separación de puntos

Cada cuánto marcar la cuerda para las alturas de trazo, medido desde el centro hacia afuera.

— de radio

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Give the span and the rise of any shallow curve — an arched opening, a curved concrete form, a bed edge, a radius wall — and the calculator returns the radius to swing it from, the length around the curve, and a list of offsets so it can be laid out with a tape and a batten when a string is not practical.

Ejemplo resuelto

A 48 in opening with an 8 in rise. Radius = (24² + 8²) ÷ (2 × 8) = 40 in. The pivot sits 40 − 8 = 32 in below the chord's centre. The half-angle is asin(24 ÷ 40) = 36.87°, so the whole arc spans 73.7° and its length is 40 × 1.287 = 51.5 in. At 12 in from the centre the offset is √(40² − 12²) − 32 = 6.16 in — 6 3/16 on a tape.

Pruebe un ejercicio

Quick referencefor arcs

Radio: (half-span² + rise²) ÷ (2 × rise)
Pivot: radius − rise back from the chord, on the centre line
Offset at x: √(radius² − x²) − (radius − rise), x from the centre
Arc length: radius × angle in radians · angle = 4 × atan(2 × rise ÷ span)

La fórmula

R = (c²/4 + h²) ÷ 2h · arc = R × 4·atan(2h ÷ c) · offset(x) = √(R² − x²) − (R − h)

c is the span and h the rise. The offsets are the height of the circle above the chord at each station; at the centre it is the rise, at the ends zero.

Preguntas

How do I find the radius of an arch from its width and height?: Radius = (half the span squared + the rise squared) ÷ (2 × the rise). A 48-inch opening with an 8-inch rise: (24² + 8²) ÷ 16 = 640 ÷ 16 = 40 inches. Swing a 40-inch string or trammel from a point 32 inches below the springing line (40 − 8) and it draws the arch.
How do I lay out an arch without a string?: Snap the chord (the straight line across the bottom), mark stations along it from the centre, and measure up from each station the offset the calculator gives. Join the points with a flexible batten. It is how you lay out an arch on a sheet of plywood, a curved form, or a flower-bed edge too big for a string.
How long is the curved edge?: Arc length = radius × the angle in radians. The calculator gives it, which is what you need for the trim, the flexible track or the edging along the curve. For the 48 × 8 in arch it is about 51.5 inches.
What if the rise is more than half the span?: Then the curve is more than a half circle — the arch bulges wider than the opening. The formula still works and the offsets are still measured up from the chord.

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