Calculate the latus rectum of a parabola, ellipse, or hyperbola. Find focus, directrix, eccentricity, and key relationships between conic section parameters.
The latus rectum is one of the most important yet often overlooked measurements of a conic section. It is the chord through a focus that is perpendicular to the principal axis of the curve. For a parabola y² = 4ax, the latus rectum has length 4a. For an ellipse or hyperbola, it equals 2b²/a. This single measurement encodes the "width" of the conic at its focus and connects eccentricity, semi-axes, and focal distance in one elegant quantity.
Understanding the latus rectum is essential in orbital mechanics (where it determines the shape of planetary orbits), optics (parabolic reflector design), and pure conic geometry. The semi-latus rectum — half the latus rectum — appears directly in the polar equation of any conic: r = ℓ/(1 + e cos θ), where ℓ is the semi-latus rectum and e is eccentricity.
This calculator supports all three main conics: parabola, ellipse, and hyperbola. Enter the defining parameters and instantly get the latus rectum, semi-latus rectum, eccentricity, foci, directrix, and a visual eccentricity gauge. Presets for real-world conic sections — satellite dishes, planetary orbits, navigation systems — let you explore applied mathematics.
The latus rectum connects multiple conic properties — foci, directrix, eccentricity, and axis lengths — in a single measurement. Calculating it by hand requires careful formula selection depending on the conic type, and mistakes are common (especially confusing a and b for ellipses vs. hyperbolas).
This calculator handles all three conic types with clear visual feedback, saving time for students working through analytic geometry, engineers designing reflective surfaces, and anyone studying orbital mechanics or optics.
Parabola (y²=4ax): Latus Rectum = 4a, Semi-LR = 2a. Ellipse (x²/a²+y²/b²=1): LR = 2b²/a, e = c/a where c = √(a²−b²). Hyperbola (x²/a²−y²/b²=1): LR = 2b²/a, e = c/a where c = √(a²+b²). Polar form: r = ℓ/(1 + e cos θ) where ℓ = semi-latus rectum.
Result: Latus Rectum = 7.20, Semi-LR = 3.60, e ≈ 0.80, Foci at (±8, 0)
For an ellipse with a = 10, b = 6: c = √(100 − 36) = 8. LR = 2(36)/10 = 7.20. Eccentricity e = 8/10 = 0.80. Foci at (±8, 0). Directrix at x = ±12.5.
In celestial mechanics, every orbit is a conic section with the central body at one focus. The semi-latus rectum ℓ is one of the six classical orbital elements and determines the size of the orbit independently of its shape. It relates to the specific angular momentum h and gravitational parameter μ via ℓ = h²/μ. For Earth orbiting the Sun, ℓ ≈ 149.45 million km — slightly less than 1 AU because Earth's orbit is slightly elliptical (e ≈ 0.0167).
Parabolic mirrors and antennas work because all rays parallel to the axis reflect through the focus. The latus rectum determines the diameter of the reflector at the focal plane, which directly affects the concentration of energy. A satellite dish with a shorter focal length (smaller a) has a shorter latus rectum, meaning a "deeper" dish shape, while a larger focal length produces a shallower profile.
By keeping the semi-latus rectum constant and varying eccentricity, you can morph smoothly between conics: a circle (e = 0), ellipse (0 < e < 1), parabola (e = 1), and hyperbola (e > 1). This is the geometric insight behind the polar equation r = ℓ/(1 + e cos θ), which unifies all conic sections into a single formula parameterized by ℓ and e. The latus rectum is thus the natural "scale parameter" of any conic section.
The latus rectum is the chord through a focus drawn perpendicular to the principal axis. Its length characterizes the "width" of the conic at the focus.
The semi-latus rectum (ℓ) is half the latus rectum. It appears in the polar equation of conics: r = ℓ/(1 + e cos θ).
For any conic, the semi-latus rectum ℓ = a(1 − e²) for an ellipse and ℓ = a(e² − 1) for a hyperbola. For a parabola (e = 1), ℓ = 2a.
In Kepler orbits, the semi-latus rectum determines the shape and size of the orbit independently of the eccentricity. It appears in the vis-viva equation and orbital energy formulas.
As e → 0, the ellipse becomes a circle and the latus rectum approaches the diameter (2a = 2b). The latus rectum equals the diameter when the conic is a circle.
For an ellipse, no — the latus rectum LR = 2b²/a is always ≤ 2b ≤ 2a. For a hyperbola, the latus rectum can exceed 2a when b > a.