Elliptical Segment Calculator

Name: Elliptical Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at an straight elliptical segment, a part of a ellipse, which is cut off by a straight line parallel to semi-axis b. b can be the longer or the shorter semi-axis.
Enter the length of semi-axis a and the height h of the cutting line, as well as the length of the semi-axis b or the area. Choose the number of decimal places, then click Calculate.

Formulas:

c = 2 a

d = 2 b

A = \frac{c d}{4} \cdot [\arccos (1 - \frac{2}{h}) - (1 - \frac{2}{h}) \cdot \sqrt{\frac{4}{h} - \frac{4}{h^{2}}}]

c and d are the two axes of the ellipse.

s = 2 b \cdot \sqrt{1 - \frac{{(a - h)}^{2}}{a^{2}}}

Axes and height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The length of the chord s can be calculated exactly, but that of the arc cannot. Therefore, the entire circumference, which would be the length of the chord plus the length of the arc, cannot be calculated exactly. Perhaps an approximate formula for the arc length could be derived from Ramanujan's approximate formula for the circumference of an ellipse.

A segment is the area between two parallel lines, whereby here one line lies outside the ellipse and is therefore irrelevant. The other line lies inside the ellipse, on the same side of the center as the outside line. Both lines are parallel to b. The chord s lies on the inner parallel. If this should be parallel to a, simply swap the values of a and b.
Such an ellipse segment is axially symmetric about the semi-axis a or the axis c, which bisects the chord s. The height h also lies on this axis. This shape has no other symmetries.
The three-dimensional equivalent of an ellipse segment would be an ellipsoid segment or a spheroid segment, i.e., the corresponding segments of an ellipsoid and a spheroid.

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