Radius of an Arc from its Chord and Height

Finds the radius of the circle an arc came from, using the straight line joining its ends (the chord) and the height from that line to the arc (the sagitta). The centre never has to be reached; measuring a piece of the arc is enough. The central angle and the arc length are also shown.

To find the radius of a curved board or a length of rail, the centre of the circle is nowhere within reach. It does not need to be. Measuring a piece of the arc is enough.

Two measurements are required: the straight line joining the ends of the arc, called the chord, and the height from the middle of that line up to the arc, called the sagitta.

r = \dfrac{\dfrac{c^2}{4} + h^2}{2h}

$r$ is the radius
$c$ is the chord
$h$ is the height of the arc above the chord

The derivation needs nothing but the Pythagorean theorem. The centre lies $r - h$ away from the chord, and joining the centre, the midpoint of the chord and one end of it forms a right triangle, giving $\left(\dfrac{c}{2}\right)^2 + (r - h)^2 = r^2$ . Expanding the left side cancels $r^2$ from both sides, leaving an equation that solves for $r$ .

Example

Take the defaults: a chord of 8 and a height of 2.

$\dfrac{8^2}{4}$ is 16 and $2^2$ is 4, so the numerator is 20. The denominator is $2 \times 2$ , or 4. The radius is exactly 5.

The central angle follows from $2\arccos\dfrac{r - h}{r}$ . Since $\arccos(3 \div 5)$ is about 53.13 degrees, the central angle is about 106.26 degrees. The arc length is about 9.2730 and the segment area about 11.1824.

The height can exceed the radius. Measuring a chord of 8 with a height of 6 gives a radius of about 4.3333, making $r - h$ negative. The formula still holds, returning a central angle of about 225.24 degrees, correctly identifying an arc larger than a semicircle. No special case is needed.

Where it is used

Woodworkers use it to reproduce an arch, surveyors to measure the radius of a road or railway curve, and opticians to check the curvature of a lens or mirror. In each case the centre is out of reach.

A longer chord produces a larger, more easily measured height. Measuring over a short span gives a small height, where a slight error throws the radius far off. This is because $h$ sits in the denominator: halve it and the radius roughly doubles. The gentler the curve, the longer the span you should measure over.

Points to watch

The height must be greater than zero. At zero the shape is a straight line, not an arc, and the radius is infinite.

Measure the height at the middle of the chord. Measuring anywhere else gives a different number.

Going the other way, when the radius is known and the arc has to be drawn, the height follows from $h = r - \sqrt{r^2 - \dfrac{c^2}{4}}$ . With a radius of 5 and a chord of 8 that is $5 - \sqrt{25 - 16}$ , or 2. This is the form to use when marking out a template.