Calculate the latus rectum of an ellipse, parabola, or conic section.
Last updated: April 2026 | By Patchworkr Team
The latus rectum is a fundamental geometric property of conic sections—specifically, it is the chord (straight line segment) that passes through a focus of the conic and is perpendicular to the major axis. The term comes from Latin: “latus” meaning side, and “rectum” meaning straight or right. For an ellipse with semi-major axis a and semi-minor axis b, the latus rectum is calculated as LR = 2b²/a. This property is ubiquitous in celestial mechanics, planetary orbits, and satellite trajectories, where the latus rectum directly relates to orbital speed and the shape of the trajectory at the focus.
The latus rectum has profound significance in mathematics and physics. In orbital mechanics, it defines the distance from the focus (where the central body is located) to the point on the orbit where the object has maximum speed. The length of the latus rectum affects the orbital eccentricity relation and appears in parametric equations of conics. For a parabola, the latus rectum equals 4p where p is the focal parameter. For a hyperbola, it is 2b²/a, similar to the ellipse. Understanding the latus rectum is essential for astronomers, engineers designing satellite orbits, and mathematicians analyzing conic section properties.
Determine whether you’re working with an ellipse, parabola, or hyperbola. Different conic types have different formulas, though this calculator uses the ellipse formula LR = 2b²/a.
Why: Each conic type has distinct mathematical properties. Correctly identifying the conic ensures you apply the right formula and get meaningful results.
Measure or identify the semi-major axis, which is half the length of the longest diameter of the ellipse. This is the distance from the center to the furthest point on the ellipse.
Why: The semi-major axis is a fundamental parameter in the latus rectum formula. It directly affects the length of the latus rectum and is essential for accurate orbital mechanics calculations.
Identify the semi-minor axis, which is half the length of the shortest diameter. This is perpendicular to the semi-major axis and is always less than or equal to a.
Why: The semi-minor axis squared is the numerator in the latus rectum formula. It characterizes the ellipse’s shape and directly influences focal chord length.
Calculate LR = 2b²/a. Square the semi-minor axis, multiply by 2, and divide by the semi-major axis. This gives the length of the latus rectum.
Why: This formula is derived from the geometric properties of conics. It gives the precise distance from the focus to the opposite side of the ellipse along a perpendicular chord.
The result is the length of the chord passing through the focus, perpendicular to the major axis. This value is always positive and helps define the orbital characteristics.
Why: Interpreting the result in context validates your calculation. High latus rectum values indicate a wider, more circular orbit, while low values suggest an elongated, elliptical path.
Calculating Latus Rectum for an Elliptical Orbit
The latus rectum is 3.6 astronomical units at the orbital focus
Semi-latus rectum is half the latus rectum (p = LR/2).
It's perpendicular (right angle) to the major axis of the conic.
In orbital mechanics, latus rectum relates to orbital shape and speed.
Yes, it's a length measurement, always positive.
Yes, for a circle where a = b, LR = 2a.
LR = a(1-e²) where e is eccentricity.
For parabola, LR = 4p where p is distance from vertex to focus.
Yes, it appears in parametric equations of conics.