Arc Radius Calculator (Radius From Chord and Rise, With Layout)

Arched aprons, curved headboards, arch-top doors and windows, bow-front drawer fronts, gentle curves on a bench seat: you usually know how wide the curve is and how high it rises, not its radius. This calculator gives the radius from those two measurements (or the rise from a radius you already have), the arc length along the curve, the angle it covers, and a list of heights at regular spacing from the centre so you can mark the curve point by point without a trammel.

Arc radius calculator

How it works

The width of the arc is a chord of a circle and the rise is its sagitta. From the circular segment formulas:

radius R = h ÷ 2 + c² ÷ (8h), where c is the chord and h the rise
rise from a radius: h = R − √(R² − c² ÷ 4)
angle of the arc θ = 2 × arcsin(c ÷ 2R); arc length = R × θ (θ in radians)
height at distance x from the centre: y = h − (R − √(R² − x²))

To draw the curve full size, measure the layout heights up from the chord line at each mark and spring a thin batten through the points; or put a nail at the centre point, R below the top of the arc, and swing a pencil on a string or a trammel stick of length R.

Worked example

A table apron 36 in long whose bottom edge should rise 3 in at the middle:

Units: Inches
Work out: Radius from width (chord) and rise
Width of the arc (chord): 36 in
Rise (height at the middle): 3 in
Layout spacing: 3 in

The calculator gives:

Radius: 55.5 in (about 55-1/2 in)
Rise at the middle: 3 in
Arc length (along the curve): 36.663 in (about 36-11/16 in)
Angle of the arc: 37.85 degrees
Trammel or string length from the centre point: 55.5 in (about 55-1/2 in)
Centre: height 3 in
3 in each side of centre: height 2.919 in (about 2-15/16 in)
6 in each side of centre: height 2.675 in (about 2-11/16 in)
9 in each side of centre: height 2.265 in (about 2-1/4 in)
12 in each side of centre: height 1.687 in (about 1-11/16 in)
15 in each side of centre: height 0.935 in (about 15/16 in)
18 in each side of centre: height 0 in

The radius is 55-1/2 in, so the trammel pivot sits 55-1/2 in below the top of the arc, well off the board; screw a scrap to the bench to carry it. The arc itself is 36-11/16 in long. If a trammel is impractical, mark the heights from the list: 2-15/16 in at 3 in from the centre, 2-11/16 in at 6 in, 2-1/4 in at 9 in, 1-11/16 in at 12 in and 15/16 in at 15 in, then bend a thin strip through the marks.

Arc radius by width and rise (inches)

Rise	24 in wide	30 in wide	36 in wide	48 in wide	60 in wide	72 in wide
1 in	72.5	113	162.5	288.5	450.5	648.5
2 in	37	57.3	82	145	226	325
3 in	25.5	39	55.5	97.5	151.5	217.5
4 in	20	30.1	42.5	74	114.5	164
6 in	15	21.8	30	51	78	111
8 in	13	18.1	24.3	40	60.3	85
12 in	12	15.4	19.5	30	43.5	60

R = h/2 + c²/8h. A dash means the rise is more than half the width (more than a half circle).

Notes and limits

The arc length tells you how long a bent lamination or a strip of edge banding must be; add a little for trimming.
Small rises give big radii. A 1 in rise over 48 in is a 288-1/2 in radius, which is easier to lay out with a bent batten than with a trammel.
For a half-round arch the rise is half the width and the radius equals the rise.
Miters and segments for polygons and round frames are in the miter angle calculator; the taper of straight legs in the taper jig calculator.

Frequently asked questions

How do you find the radius of an arc from its width and height?

Radius = height ÷ 2 + width² ÷ (8 × height). For a 36 in wide arc that rises 3 in: 1.5 + 1,296 ÷ 24 = 55.5 in.

How do I draw a large arc without a compass?

Mark the heights of the curve at regular spacing along the chord (the calculator lists them), drive small nails at the marks, and bend a thin, straight-grained strip of wood against them to trace the curve.

What is the rise of a 30 inch radius over 36 inches?

6 in. The arc is 38-5/8 in long and covers about 74 degrees.

What is the arc length of a curve?

Radius × the angle of the arc in radians. The calculator gives it directly; it is always a little longer than the straight width.

Sources

Wikipedia, "Circular segment": radius from chord and sagitta R = h/2 + c squared / 8h, chord c = 2R sin(angle/2), arc length

Last reviewed: 25 October 2026. The calculation is covered by automated tests with known answers taken from the sources above. Found an error? Please let us know.