Elliptical Segment Area Calculator

Find the area of an elliptical segment cut off by a vertical chord from the semi-major and semi-minor axes and cut position.

Input

Enter the semi-major axis a, the semi-minor axis b, and the vertical cut position x. The area of the segment on the a side of x is calculated.

Semi-major axis a

Semi-minor axis b

Cut position x

Horizontal position with the center at 0

Segment area

33.55061

Total ellipse area

125.663706

Share of total

26.7 %

Chord length

9.270248

Length uses the same unit as the inputs, and area uses that unit squared.

This finds the area of the segment of the ellipse x^2/a^2 + y^2/b^2 = 1 lying on the semi-major a side of the vertical line x.
Using the normalized value u = x / a, the segment area equals a×b×(arccos(u) − u×√(1 − u^2)).
The total area of the ellipse is π×a×b. The share is the segment area divided by the total area.
The chord length is 2×b×√(1 − u^2) in the vertical direction, longer when the cut is near the center.
Length uses the same unit as the inputs, and area uses that unit squared.

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