How this instrument works
For a parabola in the standard form y² = 4ax, the latus rectum is the chord that passes through the focus and runs perpendicular to the axis, connecting the two points where that vertical line crosses the curve. Substitute x = a — the focus sits at (a, 0) — back into y² = 4ax and the y-term becomes y² = 4a², so y = ±2a. The chord therefore runs from (a, 2a) down to (a, −2a), a straight vertical run of exactly 4a. Nothing about that derivation depends on where along the arms you look; it falls out purely from plugging the focus's own x-coordinate back into the defining equation.
The name is a fossil from classical Greek geometry: Apollonius of Perga described conics using a fixed chord at the focus centuries before x-y coordinates existed, calling it the 'straight side' — latus rectum in the Latin translations that carried the term forward. That chord, together with the vertex, was one of the tools ancient geometers used to characterize a parabola's shape without any algebra at all, which is why the length still gets its own name today rather than being folded into a generic 'width' label.
Push a toward zero and the two branches pull in tight against the axis, so the parabola opens barely at all and 4a shrinks to match; push a upward and the curve fans out fast, with the focal chord lengthening in exact step, since the relationship is a straight proportion with no bend of its own. a itself carries a second meaning worth knowing: it is also the distance from the vertex to the focus, and separately the distance from the focus to the directrix, so 4a — four of that one focal distance — is a compact way to size a parabola's opening without sketching it.
- Enter the coefficient from your equation's standard form into the field labeled 'a, in y² = 4ax'.
- Read Latus rectum length — the chord's full span, returned instantly as 4a.
- If your equation isn't already in that form, divide through until y² stands alone on one side with a plain number times x on the other, then divide that number by 4 to get a.
- Compare parabolas by entering each one's a in turn; whichever gives the larger Latus rectum length is the one that opens wider at the focus.
Worked example — a parabola with a = 3
Set a = 3 in the field labeled 'a, in y² = 4ax' and the parabola becomes y² = 12x, with its focus sitting at (3, 0) on the axis. The latus rectum is the vertical chord through that focus: substituting x = 3 back into y² = 12x gives y² = 36, so y = ±6, and the chord runs from (3, 6) to (3, −6). Latus rectum length reads 12 — exactly the distance between those two points, and exactly 4 × 3 as the formula promises, with nothing left to round.
Scale a up or down and the chord scales in exact proportion: the plainest reference parabola, y² = 4x with a = 1, has a latus rectum of just 4, while a = 10 stretches the curve to y² = 40x and a latus rectum of 40 — ten times wider at the focus for ten times the value of a, since the relationship is perfectly linear.
Questions
What is the latus rectum of a parabola?
It is the chord through the focus, drawn perpendicular to the axis, connecting the two points where that line crosses the curve. For y² = 4ax the two points work out to (a, 2a) and (a, −2a), so the chord's length is 4a — a single number that tells you how wide the parabola is at the level of its own focus.
How is the formula length = 4a derived?
Put x = a — the focus's x-coordinate — into the defining equation y² = 4ax and it becomes y² = 4a², so y = ±2a. The two intersection points are (a, 2a) and (a, −2a), a vertical separation of 2a − (−2a) = 4a. The whole derivation is one substitution; no calculus or extra geometry is needed.
Why is it called the 'latus rectum'?
Latin for 'straight side'. Apollonius of Perga used a fixed chord at the focus to characterize conic sections long before algebraic coordinates existed, and the Latin name for that chord survived translation into modern textbooks, which is why the length still carries its own dedicated term rather than a generic description.
Do ellipses and hyperbolas have a latus rectum too?
Yes, each focus of an ellipse or hyperbola has its own focal chord, with length 2b²/a rather than 4a — a different formula because those curves have a second axis length, b, that a parabola lacks. This calculator covers only the parabola's simpler case, where a single coefficient in y² = 4ax fixes the whole shape.
What's a common mistake when reading a off an equation?
Confusing this a with the leading coefficient in y = ax² + bx + c, a completely different number describing a differently oriented parabola. The a this calculator wants is the one multiplying x on the right of y² = 4ax; rearrange your equation into that exact form first, rather than lifting a coefficient from a different layout.
Where does the latus rectum matter outside pure geometry?
Parabolic reflectors — satellite dishes, telescope mirrors, car headlight bowls — all bounce parallel rays through a single focus, and the latus rectum gives the width of the reflecting surface at that focal plane, a quick check on how deep a dish needs to be for a given rim width before any detailed optical design begins.