Advanced Geometry Calculators
This category presents calculators that work with curved lines, circular components and other advanced geometric structures. Each tool provides numerical outputs describing distances, areas, angles and relationships within shapes that involve arcs, sectors and curved boundaries.
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Catenary Curve Calculator
A catenary curve forms when a uniform flexible line hangs under its own weight, creating a smooth shape defined by its span and lowest point.
Example use: estimating the curve formed by a skipping rope hanging loosely between two hands.
Inputs: horizontal span, vertical sag
Outputs: catenary constant, cable length, curvature at the lowest point, curvature at the supports, directrix distance, tension ratios, support angle, sag-to-span ratio, length-to-span ratio, sag-to-length ratio, weight ratio, area between the curve and the straight line, area beneath the curve
Visual: a hanging curve showing the span, the lowest point, and the overall shape
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Chord Length Calculator
A chord in a circle is defined by two points on the circumference, creating a straight line whose length depends on the circle's size and the arc above it.
Example use: working out the width of a circular slice when the depth of the cut is known.
Inputs: radius, diameter, central angle, distance from centre, sagitta
Outputs: chord length, central angle, apothem, sagitta, arc length, minor and major segment areas, segment perimeter, chord-to-radius ratio, arc-to-chord ratio, fraction of the full circle
Visual: a circle showing the chord, the arc above it, and the radii forming the central angle
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Cycloid Curve Calculator
A cycloid curve is traced by a point on a rolling circle, producing a repeating pattern of rises, falls, and turning points.
Example use: tracking the motion of a small mark on a rolling toy wheel.
Inputs: radius of the rolling circle, rotation angle
Outputs: horizontal and vertical position, arc length, area under the curve, radius of curvature, tangent angle, speed factor, rolling distance, slope of the tangent, normal angle, distance to the next cusp, instantaneous centre of rotation, parametric slope angle
Visual: a rolling curve with a highlighted point showing its current position and slope
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Logarithmic Spiral Calculator
A logarithmic spiral expands steadily as it rotates, with each turn increasing its distance from the centre by a constant factor.
Example use: examining how a spiral drawn from a central point grows as more turns are added.
Inputs: initial radius, growth factor, angle
Outputs: current radius, arc length, curvature, pitch angle, tangential angle, swept area
Visual: a spiral showing its radius line, turning direction, and the angle of the tangent
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Sagitta Calculator
The sagitta measures how far an arc rises above the straight chord beneath it, defining the depth of a circular segment.
Example use: checking how much a slightly bowed stick curves above the line between its ends.
Inputs: radius, chord length
Outputs: sagitta, arc length, central angle, apothem, segment area, segment perimeter, arc-to-chord ratio, sagitta-to-chord ratio, percentage of the full circle
Visual: an arc with its chord, sagitta height, and radii forming the central angle
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Superellipse (Lame Curve) Calculator
A superellipse changes shape as its exponent varies, shifting smoothly between an ellipse-like outline and a more angular form.
Example use: comparing how different rounded shapes look when adjusting the sharpness of their edges.
Inputs: horizontal semi-axis, vertical semi-axis, exponent
Outputs: area, perimeter, shape classification, rectangularity, area ratio
Visual: a superellipse shown alongside related shapes to highlight differences in roundness and outline
Advanced Geometry Calculators FAQs
They generate numerical values for curved lines, circular components and shapes defined by specific geometric relationships.
The tools cover curved paths, circular segments, arcs, spirals and other forms described by standard geometry expressions.
Results appear as numerical values describing distances, areas, angles and related measurements for curved and circular shapes.
Each calculator applies standard geometry formulas, including expressions for arc length, chord length and other curved-shape relationships.