Latus Rectum Calculator (parabola, ellipse & hyperbola)

Understanding the latus rectum calculator — concept and principle

This calculator finds the latus rectum of a parabola, ellipse or hyperbola: the chord through a focus perpendicular to the main axis. It also gives the semi-latus rectum, the eccentricity, the focal distance c and the endpoints of the latus rectum. Use it for conic sections in precalculus and for orbit geometry.

The latus rectum is the chord through a focus, perpendicular to the major axis.

How it is calculated

Parabola y² = 4px: latus rectum |4p|, endpoints (p, ±2p), e = 1.
Ellipse x²/a² + y²/b² = 1 (a ≥ b): c² = a² − b², latus rectum 2b²/a, e = c/a.
Hyperbola x²/a² − y²/b² = 1: c² = a² + b², latus rectum 2b²/a, e = c/a.
Semi-latus rectum ℓ = half the latus rectum.

The principle

The latus rectum measures how wide a conic is at its focus. For the ellipse and hyperbola, substituting x = c, the focus position, into the equation gives y = ±b²/a, so the chord is 2b²/a long. The semi-latus rectum ℓ appears in the polar equation of every conic, r = ℓ ÷ (1 + e cos θ), which is why it matters in orbital mechanics: planetary orbits are ellipses with the Sun at a focus.

Worked example

Default inputs: ellipse, a = 5, b = 3.

c = √(25 − 9) = 4, so the foci are at (±4, 0).
Eccentricity e = 4 ÷ 5 = 0.8.
Latus rectum = 2 × 3² ÷ 5 = 18 ÷ 5 = 3.6.
Semi-latus rectum ℓ = 1.8.
Endpoints: (4, ±1.8).

How to read the result

A latus rectum that is small compared with 2a means a flattened ellipse with the foci far from the centre; as e approaches 0, the ellipse becomes a circle and the latus rectum approaches the diameter 2a. For a parabola, the latus rectum is four times the focal distance.

Common mistakes

Using c² = a² + b² for an ellipse or c² = a² − b² for a hyperbola.
Swapping a and b when the major axis is vertical.
Confusing the latus rectum with the minor axis.
Using 2p instead of 4p for a parabola.

Keep in mind: The formulas use the standard position with the main axis along x. For rotated or shifted conics, transform to standard form first.

Frequently asked questions

What is the latus rectum of an ellipse?

The chord through a focus perpendicular to the major axis, with length 2b²/a.

What is the latus rectum of y² = 8x?

4p = 8, since 4p = 8 gives p = 2.

Why is the semi-latus rectum important?

It appears in the polar equation of conics used to describe orbits.

Sources: Lumen Learning, Precalculus (based on OpenStax Precalculus 2e), "Conics: Cheat Sheet"

Checked 9 Nov 2026. Lumen Learning, Precalculus (based on OpenStax Precalculus 2e), “Conics: Cheat Sheet” (ellipse foci c² = a² − b², hyperbola foci c² = a² + b², parabola focus p units from the vertex, latus rectum length |4p|, eccentricity e = PF/PD). The 2b²/a values follow by substituting x = c into the standard equations.

Your numbers