What this calculator does
The latus rectum of a conic section is the chord that passes through a focus at right angles to the major axis. It is a single number that describes how wide the curve opens at the focus, which is why it appears throughout the standard equations for parabolas, ellipses and hyperbolas.
For a parabola written as y² = 4ax the latus rectum is simply 4a. For an ellipse and a hyperbola it is 2b² divided by a, using the semi-major axis a and the semi-minor axis b. This calculator applies whichever of those applies to the curve you select.
The formula
For a parabola, multiply the focal distance a by 4. For an ellipse or hyperbola, square the semi-minor axis b, double it, and divide by the semi-major axis a. Half of the result is the semi-latus rectum, which is the value that appears in the polar form of a conic.
| Term | Meaning |
|---|---|
| Latus rectum | The chord through a focus, perpendicular to the major axis, running from one side of the curve to the other. |
| Semi-latus rectum | Half the latus rectum, usually written as l, and the value used in the polar equation of a conic. |
| Eccentricity | How far the conic departs from a circle: 0 for a circle, between 0 and 1 for an ellipse, and greater than 1 for a hyperbola. |
The inputs explained
| Field | What to enter |
|---|---|
| Conic section | Choose the type of conic section. The formula changes between a parabola and the other two. |
| a (parabola: focal distance; ellipse/hyperbola: semi-major axis) | For a parabola, the distance from the vertex to the focus in y² = 4ax. For an ellipse or hyperbola, the semi-major axis. |
| b (semi-minor axis, ellipse and hyperbola only) | The semi-minor axis, used only for an ellipse or a hyperbola. |
When to use it
Working through a conic sections problem
Latus rectum questions are standard in analytic geometry courses, and checking a hand-worked answer against the formula is a quick way to confirm the right one was applied.
Describing an orbit
Orbits are conics with the primary body at one focus, and the semi-latus rectum is the parameter that appears directly in the polar form of the orbit equation.
Setting out a parabolic shape
For a parabolic reflector or arch, the latus rectum fixes the width of the curve at the focus, which is often the practical dimension being designed around.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the latus rectum of a parabola change with a?
The focal distance a, varied, for a parabola in standard form.
| a (focal distance) | Latus rectum | Semi-latus rectum |
|---|---|---|
| 1 | 4.000 | 2.000 |
| 2 | 8.000 | 4.000 |
| 3 | 12.000 | 6.000 |
| 4 | 16.000 | 8.000 |
| 5 | 20.000 | 10.000 |
Questions
Why is the parabola formula different?
A parabola has no second axis in the way an ellipse and hyperbola do, so there is no b to work with. Writing it as y² = 4ax builds the focal distance directly into the equation, and the latus rectum falls out as 4a.
What is the semi-latus rectum used for?
It is the standard parameter in the polar equation of a conic, which is the form used in orbital mechanics. Writing an orbit in that form makes the semi-latus rectum a direct input rather than something derived.
Does the ellipse formula work if b is larger than a?
By convention a is the semi-major axis, so it should be the larger of the two for an ellipse. If b exceeds a the labels have been swapped, and the eccentricity cannot be calculated, which this calculator reports rather than returning a misleading figure.
Is the latus rectum the same as the focal width?
Yes, focal width is another name for the same chord. Both refer to the width of the curve measured through the focus, perpendicular to the major axis.
For the other properties of an ellipse, see the ellipse calculator. For the roots and vertex of a parabola written as a quadratic, see the quadratic equation calculator.