Latus Rectum Calculator
Calculate the full and semi latus rectum lengths of parabolas, ellipses, and hyperbolas using the conic type, parameter, and eccentricity. Latus Rectum Calculator helps students, engineers, and mathematicians analyze conic sections and coordinate geometry problems.
Highlights
- Calculate full and semi-latus rectum lengths instantly.
- Supports parabolas, ellipses, and hyperbolas.
- Uses the primary parameter and eccentricity for accurate results.
- Works with multiple metric and imperial units.
- Ideal for students, engineers, and coordinate geometry problems.
Table of Contents
How Latus Rectum Calculator Works?
The latus rectum is the focal chord of a conic section: a chord that passes through a focus and is perpendicular to the major axis, transverse axis, or axis of symmetry. The calculator uses the primary parameter a and eccentricity e to determine both the full latus rectum and the semi-latus rectum.
For a parabola, the eccentricity is e = 1, and the supplied parameter a is treated as the focal parameter. For ellipses, a is the semi-major axis and the eccentricity satisfies 0 < e < 1. For hyperbolas, a is the semi-transverse axis and e > 1.
Here, p is the semi-latus rectum, which is half of the complete focal chord. Therefore, the full latus rectum is twice the semi-latus rectum.
| Conic | Eccentricity condition | Semi-latus rectum | Full latus rectum |
|---|---|---|---|
| Parabola | e = 1 | a | 2a |
| Ellipse | 0 < e < 1 | a(1 − e2) | 2a(1 − e2) |
| Hyperbola | e > 1 | a(e2 − 1) | 2a(e2 − 1) |
For example, if an ellipse has a = 5 and e = 0.6, its semi-latus rectum is 5(1 − 0.62) = 3.2, and its full latus rectum is 6.4 in the same length unit as a.
Practical Use Cases
Students can use the calculator to check conic-section homework involving parabolas, ellipses, and hyperbolas, especially when converting between eccentricity-based formulas and focal chord properties.
Geometry teachers can demonstrate how changing eccentricity affects the latus rectum, showing that an ellipse's latus rectum decreases as its eccentricity approaches 1 while a hyperbola's increases as eccentricity grows.
Engineers and designers can calculate focal chord dimensions when analyzing reflector profiles, orbital cross-sections, curved structures, or other designs based on conic geometry.
Mathematicians and researchers can quickly obtain the semi-latus rectum used in polar equations of conics and in geometric relationships involving focus, directrix, and eccentricity.
Astronomy and aerospace users can use the hyperbola or ellipse formulas as part of preliminary orbit and trajectory analysis, where the semi-latus rectum is a standard conic parameter.
Accuracy and Limitations
The calculator evaluates the exact conic formulas from the entered values and reports the full latus rectum as twice the semi-latus rectum, subject to normal numerical rounding in the displayed result. The parameter a must be positive and within the supported range of 0.0001 to 1,000,000. Results are mathematically meaningful only when eccentricity matches the selected conic: a parabola requires e = 1, an ellipse requires 0 < e < 1, and a hyperbola requires e > 1. Values that violate these conditions may produce a formula result that does not represent the selected conic.
Quick Examples
Example #1: Standard Parabola
Calculates the full and semi-latus rectum lengths for a parabola with a focal parameter of 5 meters.
Example #2: Wide Ellipse
Determines the latus rectum lengths for an ellipse with a 12-centimeter semi-major axis and eccentricity of 0.8.
Example #3: Hyperbolic Orbit
Calculates the full and semi-latus rectum lengths for a hyperbola with a 3-kilometer primary parameter and eccentricity of 1.5.
