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Latus Rectum Calculator

Calculate the length and endpoints of the latus rectum for a parabola, ellipse, or hyperbola.

Reviewed by Christopher FloiedPublished Updated

This free online latus rectum calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Leading coefficient of the parabola y = ax² + bx + c

x coordinate of the vertex

y coordinate of the vertex

Results

Latus Rectum Length

1

Focus y

0.25

Left Endpoint x

-0.5

Right Endpoint x

0.5

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Latus Rectum Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Latus Rectum Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Latus Rectum Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Latus Rectum Calculator

The latus rectum is a chord of a conic section that passes through a focus and is perpendicular to the major axis (or axis of symmetry). For a parabola y = ax², the latus rectum has length |1/a| = 4|p|, where p is the distance from the vertex to the focus. The latus rectum is a key measurement because it determines the width of the conic at the focus level, providing a natural scale for the curve's openness. In optics, the latus rectum of a parabolic mirror determines the size of the focal spot and the field of view. For ellipses, the latus rectum length is 2b²/a, where a and b are the semi-major and semi-minor axes. For hyperbolas, the formula is the same: 2b²/a. The concept unifies all conic sections through the focus-directrix property: any conic can be defined as the locus of points whose distance to a focus divided by distance to a directrix equals the eccentricity, and the latus rectum connects these quantities. This calculator computes the latus rectum for a parabola given its vertex and leading coefficient.

The Math Behind It

For a parabola y = a(x - h)² + k, the focal parameter is p = 1/(4a), meaning the focus is at (h, k + p) and the directrix is y = k - p. The latus rectum is the horizontal chord through the focus, perpendicular to the axis of symmetry x = h. Its endpoints are at (h - 1/(2a), k + p) and (h + 1/(2a), k + p), giving a total length of |1/a|. Since 1/a = 4p, the latus rectum length equals four times the focal distance. A larger |a| means a narrower parabola with a shorter latus rectum. For an ellipse with semi-axes a and b (a > b), the latus rectum length is 2b²/a, and for a hyperbola it is 2b²/a. In the focus-directrix definition of conics, every conic satisfies r = l/(1 + e cos(θ)) in polar coordinates centered at a focus, where l is the semi-latus rectum (half the latus rectum) and e is the eccentricity. This polar equation unifies circles (e = 0), ellipses (0 < e < 1), parabolas (e = 1), and hyperbolas (e > 1). The semi-latus rectum l is related to the full latus rectum L by L = 2l, and for an ellipse l = b²/a = a(1 - e²).

Formula Reference

Latus Rectum (Parabola)

L = |1/a| = 4|p|, where p = 1/(4a)

Variables: a = leading coefficient; p = focal parameter

Latus Rectum (Ellipse)

L = 2b²/a

Variables: a = semi-major axis; b = semi-minor axis

Worked Examples

Example 1: Latus rectum of y = x²

Find the latus rectum of the parabola y = x² (a = 1, vertex at origin).

Step 1:Latus rectum length = |1/1| = 1
Step 2:Focus y = 0 + 1/(4×1) = 0.25
Step 3:Left endpoint: x = 0 - 1/2 = -0.5
Step 4:Right endpoint: x = 0 + 1/2 = 0.5

The latus rectum has length 1, with endpoints (-0.5, 0.25) and (0.5, 0.25).

Example 2: Latus rectum of y = 0.25x²

Find the latus rectum of y = 0.25x² (a = 0.25, vertex at origin).

Step 1:Latus rectum = |1/0.25| = 4
Step 2:Focus y = 1/(4 × 0.25) = 1
Step 3:Left: x = -2, Right: x = 2

The latus rectum is 4 units long, from (-2, 1) to (2, 1). The wider parabola has a longer latus rectum.

Common Mistakes & Tips

  • !Confusing the latus rectum with the directrix — the latus rectum is a chord through the focus, not a line.
  • !Using the wrong formula for different conics — parabola uses |1/a|, ellipse uses 2b²/a.
  • !Forgetting that the latus rectum is perpendicular to the axis of symmetry, not parallel to it.
  • !Confusing semi-latus rectum (half the latus rectum) with the full latus rectum.

Related Concepts

Parabola

The conic section whose latus rectum is most commonly calculated.

Conic Sections

The latus rectum is defined for all conic sections.

Vertex Form

Vertex form directly provides the parameters needed for the latus rectum.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Conic Sections Calculator

Parabola Calculator

Frequently Asked Questions

Why is the latus rectum important?

It measures the width of the conic at the focus, determining properties like the field of view in parabolic mirrors. In the unified polar equation of conics, the semi-latus rectum is the key parameter alongside eccentricity.

What is the semi-latus rectum?

The semi-latus rectum is half the latus rectum. For a parabola with focal parameter p, the semi-latus rectum equals 2p and the full latus rectum equals 4p.

How does the latus rectum relate to the width of a parabola?

The latus rectum gives the width of the parabola at the height of the focus. A longer latus rectum means a wider parabola at the focal level. It is inversely proportional to |a|.

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