Latus Rectum Calculator — Parabola, Ellipse & Hyperbola
Select a conic section and enter its parameters to find the length of the latus rectum — the chord through a focus perpendicular to the principal axis. Works for parabolas (y = ax²), ellipses and hyperbolas.
Conic section
LR = 1/|a| = 4p for parabola y = ax²
- 1
Focal length p
p = 1 ÷ (4 × |a|) = 1 ÷ (4 × 0.25) = 1 - 2
Latus rectum
LR = 4p = 1 ÷ |a| = 1 ÷ 0.25 = 4
How does this calculator work?
Latus rectum is the conic chord through a focus, perpendicular to the principal axis. For a parabola y = ax²: LR = 1/|a|. For an ellipse or hyperbola: LR = 2b²/a, where a is the major/real semi-axis and b is the minor/conjugate semi-axis. The semi-latus rectum = LR/2 = b²/a is the orbital parameter p.
Formula
How this is calculated
The latus rectum of a conic section is the chord that passes through a focus and is perpendicular to the principal axis. Its length is a natural measure of the "width" of the conic at the focus. The semi-latus rectum (half of LR) appears in orbital mechanics as the parameter p of a Keplerian orbit.
For a parabola y = ax², the focus is at (0, p) where p = 1/(4a), and the latus rectum has length LR = 4p = 1/|a|. For an ellipse x²/a² + y²/b² = 1 (a ≥ b), the foci lie at (±c, 0) where c = √(a²−b²), and the latus rectum at each focus has length 2b²/a. For a hyperbola x²/a² − y²/b² = 1, the foci are at (±c, 0) with c = √(a²+b²), and the formula 2b²/a still applies.
A parabola has eccentricity exactly 1. An ellipse has 0 < e < 1 (a circle is the degenerate case e = 0). A hyperbola has e > 1. All three formulas share the same structure because they are all conic sections — slices of a double cone at different angles.
Frequently asked questions
In orbital mechanics, the semi-latus rectum p = b²/a (or LR/2) is the parameter in the polar equation of a Keplerian orbit: r = p / (1 + e·cos θ). It gives the orbital radius at 90° from periapsis regardless of eccentricity. It also appears in reflector antenna design and in the geometry of conic mirrors.
A focal chord is any chord that passes through a focus. The latus rectum is the specific focal chord perpendicular to the major axis (principal axis). It is the shortest focal chord for an ellipse and the unique perpendicular one for a parabola.
a is the semi-major axis (the longer one) and b is the semi-minor axis (the shorter one). If you enter a < b, the calculator automatically swaps them so the formula LR = 2b²/a uses the correct orientation.
Also known as
TG we-Calculate Editorial Team. (2026). Latus Rectum Calculator — Parabola, Ellipse & Hyperbola [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/latus-rectum-calculator
TG we-Calculate Editorial Team. "Latus Rectum Calculator — Parabola, Ellipse & Hyperbola." TG we-Calculate. 2026. https://we-calculate.com/calculator/latus-rectum-calculator.
TG we-Calculate Editorial Team, "Latus Rectum Calculator — Parabola, Ellipse & Hyperbola," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/latus-rectum-calculator
@misc{wecalculate_latus_rectum_calculator, title = {Latus Rectum Calculator — Parabola, Ellipse & Hyperbola}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/latus-rectum-calculator}}, year = {2026}, note = {TG we-Calculate} }
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