Circular Central Segment Calculator

Name: Circular Central Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at a circular central segment. This is formed by a circle, of which two circular segments are cut off. The cut egdes don't need to be parallel, but must not cross within the circle. The circular central segment may extend beyond the center of the circle, this is unlike to the calculator for the circular layer. The three-dimensional equivalent of the circular central segment is the spherical central segment.
Enter the radius of the circle and the heights of the two cut off segments. Choose the number of decimal places, then click Calculate.

Formulas:

H = 2 r - h - i

s = 2 \cdot \sqrt{2 \cdot r h - h^{2}}

t = 2 \cdot \sqrt{2 \cdot r i - i^{2}}

α = 2 \cdot \arccos (1 - \frac{h}{r})

β = 2 \cdot \arccos (1 - \frac{i}{r})

p = 2 π r - r (α + β) + s + t

A = π r^{2} - \frac{r^{2} \cdot α}{2} + \frac{s \cdot (r - h)}{2} - \frac{r^{2} \cdot β}{2} + \frac{t \cdot (r - i)}{2}

pi:

π = 3.141592653589793...

Radius, heights, lengths and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

α and β are the angles between the respective legs from the center of the circle to the straight edge of a circular segment. The perimeter is that of the circle minus the two arc lengths of the segments plus the two edge lengths of the segments. This perimeter is smaller than the circumference of the underlying circle. The volume of the central circular segment is the volume of the circle minus the volumes of the two outer circular segments.

In the sketch, the straight edges of the circular segments are parallel. However, this is irrelevant for the surface area and volume of this figure. As long as the two lines do not intersect, these values do not change simply because the cuts are different.
The circular central segment with parallel lines is axially symmetric to the line that passes centrally and perpendicularly through the two parallel lines. If the two segments have the same height, the central circular segment is even point-symmetric. These symmetry properties are lost if the cuts are shifted, meaning that the edges of the two circular segments are no longer parallel.

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