Ellipse segment area calculator

When the chord is a vertical line x equals cut_x on the standard ellipse x squared over a squared plus y squared over b squared equals one, a closed-form identity gives the area without numerical integration. This capacity accepts positive semi-axes a and b and a cutting abscissa cut_x that must satisfy the absolute value of cut_x less than or equal to a so the line actually intersects the ellipse. Cuts strictly outside the ellipse fail with invalid_input. The response returns segment_area as the smaller piece, plus area_right, area_left, total_area, chord length, and the normalized cut u equals cut_x over a.

What an elliptical segment is and how a vertical cut fixes its area

In Euclidean plane geometry an axis-aligned ellipse centered at the origin is fixed by two positive semi-axes a along x and b along y. The left area is the total area pi a b minus that right area. The classical segment is the smaller of those two pieces: when the cut sits near the right vertex the right cap is the segment; when the cut sits left of center the left cap may be smaller instead. At the vertices the segment collapses to zero area and the chord degenerates to a point. Search intent clusters around ellipse segment area, elliptical cap area, area of ellipse cut by a vertical line, and semi-axes chord calculator. A vertical chord at x equals cut_x meets the curve at two points with the same abscissa and opposite ordinates y equals plus or minus b times the square root of one minus u squared, where u is cut_x divided by a. That chord splits the elliptical disk into a left region and a right region. The area of the right region—points with x greater than or equal to cut_x—follows from integrating the vertical chord length from cut_x to a, which evaluates in closed form to a times b times the quantity arccos of u minus u times the square root of one minus u squared. At cut_x equals zero on any ellipse the two halves share area pi a b over two. Returning segment_area together with area_right, area_left, total_area, chord_half_height, and chord_length makes the payload easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing arccos and the affine stretch from the unit circle in every language you teach or deploy.

How this calculator validates input and applies the segment formula

Both semi-axes must be strictly positive, large enough for stable floating-point evaluation, and below a generous upper bound so extreme products cannot overflow intermediate terms. Chord half-height is b times that square root and chord length is twice the half-height. Send a JSON body with three required numeric fields: a, the positive semi-axis along x; b, the positive semi-axis along y; and cut_x, the abscissa of the vertical cutting line. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input naming the field. The domain rule for the cut is absolute value of cut_x less than or equal to a: any cut with absolute value strictly greater than a lies outside the ellipse and is rejected immediately with a message that names cut_x and a, so callers never receive a silent NaN from arccos of an out-of-range argument. Total area is pi a b; area_left is total minus area_right; segment_area is the minimum of the two caps. All floating results are rounded to a stable decimal precision so the same input always yields the same JSON across runtimes. Edge cases are explicit: a circle is the special case a equals b and recovers half-disk and full-disk limits correctly; a cut at either vertex returns segment_area zero with a degenerate chord; a cut outside fails before any area is returned.

Practical use, limits, pricing, and delivery channels

Use the browser tool when you want an immediate elliptical segment area while checking homework, sizing a partial elliptical tank cross-section, verifying an arch or plate cut-out, or confirming a shop drawing without installing a computer-algebra system. Length units are whatever you already use for the figure, and response lengths and areas share that same unit system—there is no unit conversion inside the capacity. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Compared with a circular segment calculator that takes radius and central angle, this tool targets the elliptical search intent with vertical-cut parameterization; compared with a full ellipse metrics tool that returns area and perimeter only, it answers the chord-cap question that design and homework workflows actually ask. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same a, b, and cut_x always yield the same segment_area, caps, and chord fields. Wire the endpoint into ellipse-segment drills, contest answer checkers, elliptical tank and pipe worksheets, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a cut beyond the semi-axis is rejected. Keep requests small, pin the rounded JSON in tests, and you get reproducible segment areas without standing up symbolic math infrastructure of your own.