To calculate the Arc Height (s):
$$s = r \pm \sqrt{r^2 - \left(\frac{L}{2}\right)^2}$$
Where:
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.
Let's assume the following values:
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: $$s = 10 + 8 = 18 \text{ inches}$$
Let's assume the following values:
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: $$s = 15 + 11.18 = 26.18 \text{ inches}$$
