Slicing a cone with a plane gives a circle, ellipse, parabola or hyperbola. Planets move on ellipses, thrown balls on parabolas and some comets on hyperbolas; satellite dishes and headlights use the focus property of parabolas.
How to use the Conic Sections Calculator
- Conic section
- Semi-major axis a (or radius)
- Semi-minor axis b
- Focal length f (vertex to focus)
Results update as you type, and everything is calculated in your browser.
How it works
Choose the conic. For ellipses and hyperbolas enter the semi-axes a and b; for a parabola enter the focal length f (vertex to focus). The eccentricity tells you the type: 0 for a circle, below 1 for an ellipse, 1 for a parabola and above 1 for a hyperbola.
Inputs explained
| Field | What to enter |
|---|---|
| Conic section | Choose one: Circle, Ellipse, Parabola, Hyperbola |
| Semi-major axis a (or radius) | A number |
| Semi-minor axis b | A number |
| Focal length f (vertex to focus) | A number |
Example values
How the main result changes when Semi-major axis a changes and everything else stays at the example values above:
| Semi-major axis a (or radius) | Eccentricity e |
|---|---|
| 3.75 | 0.6 |
| 5 | 0.8 |
| 6.25 | 0.8773 |
| 7.5 | 0.9165 |
| 10 | 0.9539 |
Formula
Ellipse: c = √(a² − b²), e = c/a · Hyperbola: c = √(a² + b²), e = c/a
Semi-latus rectum ℓ = b²/a, focal parameter p = b²/c · Parabola: e = 1, ℓ = 2f
Worked example
Inputs — Conic section: Ellipse; Semi-major axis a (or radius): 5; Semi-minor axis b: 3.
Results — Eccentricity e = 0.8; Linear eccentricity c (centre to focus) = 4; Semi-latus rectum ℓ = 1.8; Focal parameter p = 2.25.
Key terms
- Focus
- A special point used to define a conic.
- Eccentricity
- c/a, the shape parameter of a conic.
Frequently asked questions
What is eccentricity?
A number measuring how far a conic is from a circle.
Why is a circle an ellipse?
It is an ellipse with a = b, so e = 0.
What is the latus rectum?
The chord through a focus perpendicular to the major axis; ℓ is half of it.
Which conic has e = 1?
The parabola.