Arc Height (sagitta) Calculator

Enter the radius of the arc and the length of the base (chord) of the arc into the calculator to determine the arc height(s).

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Arc Height Formula

The following formula is used to calculate the height of a circular arc (sagitta) from its chord length and radius.

s = r \pm \sqrt{r^2 - (L/2)^2}

• Where s is the arc height (sagitta), measured from the chord to the arc along a line perpendicular to the chord
• r is the radius of the arc
• L is the base of the arc (chord length)

For a given chord in a circle, there are two corresponding circular segments (a minor segment and a major segment), so the formula can return a small height (using “−”) and a large height (using “+”).

Arc Height Definition

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.

How to calculate arc height?

Arc Height Example Problem:

The following example problem is a step-by-step guide on how to calculate an arc height.

First, determine the radius of the arc. In this example, the radius is measured as 5 inches.

Next, measure the base of the arc. For this problem, the chord is measured to be 7 inches.

Finally, calculate the arc height using the formula above:

s = r ± √ ( r² – (L/2)² )

s = 5 ± √ ( 5² – (7/2)² )

s = 8.5707 or 1.4293

FAQ

What is the difference between arc length and arc height?

Arc length refers to the distance measured along the curve of the arc, from one end of the arc to the other. Arc height (sagitta), on the other hand, measures the distance from the midpoint of the chord (base of the arc) to the arc along a line perpendicular to the chord.

Can the arc height be larger than the radius of the arc?

It depends on which segment you mean. For the minor segment (the “small” arc height), the arc height is always less than or equal to the radius. For the major segment (the “large” arc height), the height can be larger than the radius (and is less than or equal to 2 × radius, with 2r occurring only in the degenerate case of a zero-length chord).

Why is the arc height important in calculations and design?

The arc height is a crucial measurement in various fields such as engineering, architecture, and mathematics because it helps in determining the curvature of an arc, which is essential for designing arches, bridges, lenses, and other curved structures. Knowing the arc height can also aid in calculating the area of a segment and the positioning of elements in a circular layout.