Last updated: July 31, 2026
Latus Rectum Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Parabola: 4p. Ellipse/Hyperbola: 2b²/a. Ellipse e = √(1-(b/a)²), Hyperbola e = √(1+(b/a)²).
Where:
- p=Parabola parameter
- a=Semi-major axis
- b=Semi-minor axis or conjugate semi-axis
- LR=Latus rectum length
- e=Eccentricity
Worked Examples
Parabola example
For p = 3, the latus rectum is immediate.
- 1Use LR = 4p.
- 2LR = 4 × 3 = 12.
- 3No ellipse/hyperbola axis values are needed.
Ellipse example
Use a = 5 and b = 3.
- 1LR = 2b²/a = 18/5 = 3.6.
- 2e = √(1-(3/5)²) = √(16/25) = 0.8.
- 3Both values are returned.
Hyperbola example
Use a = 4 and b = 3.
- 1LR = 2b²/a = 18/4 = 4.5.
- 2e = √(1+(3/4)²) = √(25/16) = 1.25.
- 3Hyperbola eccentricity is always greater than 1.
Introduction
The latus rectum is a focal chord that plays a central role in parabola, ellipse, and hyperbola geometry. This calculator switches formulas by conic type and also reports eccentricity for ellipse and hyperbola cases.
Different Conic Types
Each conic uses a different set of parameters.
Parabola mode uses the parameter p and LR = 4p.
Ellipse mode uses semi-major axis a and semi-minor axis b.
Hyperbola mode also uses a and b but with a different eccentricity formula.
The calculator checks the selected type before computing.
Shared Formula for Ellipse and Hyperbola
Ellipse and hyperbola both use LR = 2b²/a.
Only the eccentricity formula changes between them.
Ellipse eccentricity is between 0 and 1.
Hyperbola eccentricity is greater than 1.
The same length formula makes comparisons easy.
Validation Rules
Each conic type has its own geometric restrictions.
Parabola requires p > 0.
Ellipse requires a > b > 0.
Hyperbola requires a > 0 and b > 0.
Unknown conic types return an error.
Why Eccentricity Matters
Eccentricity measures how far the conic departs from circular behavior.
Ellipse eccentricity approaches 0 as the shape becomes more circular.
Parabola has eccentricity 1 mathematically, though this calculator only reports latus rectum in that mode.
Hyperbola eccentricity is always greater than 1.
Combining LR and e gives a fuller picture of the conic.
Worked Examples
The sample values give exact or simple decimal answers.
Parabola p = 3 gives LR = 12.
Ellipse a = 5, b = 3 gives LR = 3.6 and e = 0.8.
Hyperbola a = 4, b = 3 gives LR = 4.5 and e = 1.25.
These results are all easy to verify manually.
Where It Is Used
Latus rectum calculations appear in both analytic geometry and applied math.
Conic section homework and exams.
Orbit and reflective-property discussions.
Coordinate geometry references.
Quick comparison of conic shapes from axis data.
FAQs
What is a latus rectum?
It is a chord through a focus and perpendicular to the principal axis of the conic.
Why does ellipse mode require a > b?
That keeps a as the semi-major axis and ensures the eccentricity formula stays valid.
Does hyperbola use the same latus rectum formula as ellipse?
Yes. Both use 2b²/a in this standard parameterization.
Why is eccentricity not emphasized for parabola mode?
The requested outputs focus on latus rectum there, while eccentricity is mainly reported for ellipse and hyperbola.
What if I enter an unknown conic type?
The calculator returns an error instead of attempting a calculation.
Can I use decimals for a, b, or p?
Yes. Any valid positive finite decimals are supported.