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Arc Height (Sagitta) Calculator

Formula

To calculate the Arc Height (s):

$s = r \pm \sqrt{r^2 - \left(\frac{L}{2}\right)^2}$

Where:

s is the arc height
r is the radius of the arc
L is the base of the arc (chord length)

What is an Arc Height?

An arc height, also commonly referred to as a sagitta, is a measure of the distance from the base of a circular segment to the peak of the arc. The distance is measured from the center of the base (chord) to the peak.

Example Calculation 1

Let's assume the following values:

Radius (r) = 10 inches
Chord Length (L) = 12 inches

Using the formula:

$s = 10 \pm \sqrt{10^2 - \left(\frac{12}{2}\right)^2} = 10 \pm \sqrt{100 - 36} = 10 \pm \sqrt{64} = 10 \pm 8$

Small Arc Height: $s = 10 - 8 = 2 \text{ inches}$

Large Arc Height: $s = 10 + 8 = 18 \text{ inches}$

Example Calculation 2

Let's assume the following values:

Radius (r) = 15 inches
Chord Length (L) = 20 inches

Using the formula:

$s = 15 \pm \sqrt{15^2 - \left(\frac{20}{2}\right)^2} = 15 \pm \sqrt{225 - 100} = 15 \pm \sqrt{125} = 15 \pm 11.18$

Small Arc Height: $s = 15 - 11.18 = 3.82 \text{ inches}$

Large Arc Height: $s = 15 + 11.18 = 26.18 \text{ inches}$