Latus Rectum Calculator - Parabola, Ellipse, Hyperbola
Find the latus rectum length for any conic section: parabola (4p), ellipse, or hyperbola (2b²/a).
Select the conic section type and enter the required parameters to compute the latus rectum length instantly.
Latus Rectum Calculator - Parabola, Ellipse, Hyperbola
Find the latus rectum length for any conic section: parabola (4p), ellipse, or hyperbola (2b²/a).
Latus Rectum Examples
Four examples across all three conic section types.
| Parameters | Latus Rectum | Conic / Formula |
|---|---|---|
| Parabola, p = 2 | 8 | Parabola: L = 4p = 4 × 2 = 8. |
| Ellipse, a = 5, b = 3 | 3.6 | Ellipse: L = 2b²/a = 2 × 9 / 5 = 3.6. |
| Hyperbola, a = 4, b = 2 | 2 | Hyperbola: L = 2b²/a = 2 × 4 / 4 = 2. |
| Parabola, p = 10 | 40 | Parabola: L = 4p = 4 × 10 = 40. |
About the Latus Rectum Calculator
The latus rectum is a special chord of a conic section that passes through a focus and is perpendicular to the principal axis. Its name comes from Latin, meaning “straight side.” The latus rectum has a different formula for each of the three primary conic sections: the parabola, the ellipse, and the hyperbola.
For a parabola described by the equation y² = 4px or x² = 4py, the length of the latus rectum is simply 4p, where p is the distance from the vertex to the focus (also called the focal parameter). The latus rectum connects the two points on the parabola that are directly above and below (or left and right of) the focus. A larger value of p means the parabola opens more gradually and has a longer latus rectum.
For an ellipse with semi-major axis a and semi-minor axis b (where a > b), the length of the latus rectum is 2b² / a. This formula applies to both the horizontal ellipse (x²/a² + y²/b² = 1) and the vertical ellipse. The latus rectum is the chord through each focus perpendicular to the major axis, and there are actually two such chords, one at each focus, both of the same length. The more elongated the ellipse (smaller b relative to a), the shorter the latus rectum.
For a hyperbola with semi-transverse axis a and semi-conjugate axis b, the same formula 2b² / a gives the length of each latus rectum. A hyperbola has two branches and two foci, so it has two latus recta, one for each branch. Despite the very different shape of a hyperbola compared to an ellipse, the formulas are identical when expressed in terms of a and b.
The latus rectum is a fundamental property used in several areas of mathematics and physics. In optics, parabolic mirrors and antennas focus parallel rays at the focal point; the latus rectum determines the width of the parabola at the focal depth, which affects the aperture of the optical system. In astronomy, the latus rectum of an elliptical orbit determines the distance from the focus (the star or planet being orbited) at which the velocity is exactly the average of the maximum and minimum orbital velocities. Kepler’s laws and orbital mechanics calculations use the latus rectum as a convenient orbital parameter.
This calculator automates the arithmetic: select the conic type, enter the appropriate parameter(s), and the tool computes the latus rectum length immediately. For a parabola you need only p. For an ellipse or hyperbola you need both a and b.
How to Use the Latus Rectum Calculator
- Select the conic section type from the dropdown: Parabola, Ellipse, or Hyperbola.
- For a Parabola, enter the value of p (the distance from the vertex to the focus). For an Ellipse or Hyperbola, enter the semi-major axis a and semi-minor axis b.
- Click “Calculate Latus Rectum” to compute the result.
- The result displays the latus rectum length along with the formula used (4p for parabola, 2b²/a for ellipse and hyperbola).
- Click Reset to clear the inputs and start a new calculation with a different conic section.
Frequently Asked Questions
What is the latus rectum of a conic section?
The latus rectum is the chord passing through a focus of the conic that is perpendicular to the principal axis. Its length is a key geometric property that characterizes the “width” of the conic at the focal level. For a parabola it equals 4p, and for an ellipse or hyperbola it equals 2b²/a.
Why does the same formula work for both the ellipse and hyperbola?
Although an ellipse and a hyperbola look very different, both are described by equations involving semi-axes a and b, and both have foci at a distance c from the center. The latus rectum length can be derived from the fundamental relation b² = a² − c² (ellipse) or b² = c² − a² (hyperbola), and in both cases the resulting formula simplifies to 2b²/a.
What is the difference between the semi-major and semi-minor axes?
For an ellipse, the semi-major axis a is half the length of the longest diameter, and the semi-minor axis b is half the length of the shortest diameter. For a hyperbola, a is the semi-transverse axis (half the distance between the vertices) and b is the semi-conjugate axis. In all cases, a must be entered as the larger of the two values for the ellipse constraint to hold.
How is the latus rectum used in astronomy?
In orbital mechanics, a planet’s orbit is an ellipse with the Sun at one focus. The semi-latus rectum (half the latus rectum length) relates the orbital geometry to physical quantities. It appears in the orbit equation r = l / (1 + e∂cosθ), where l is the semi-latus rectum and e∂ is the eccentricity. It determines the orbital radius when the true anomaly is 90°, i.e., when the planet is directly “abeam” the focus.
Can the latus rectum be used for a circle?
A circle is a special case of an ellipse with a = b and eccentricity zero. The foci both collapse to the center, and the “latus rectum” through the center has length 2a = the diameter. This calculator is designed for the general conic cases; for a circle simply note that the latus rectum equals the diameter.