# Conic Sections Calculator

URL: https://speedycalculator.com/calculators/math/conic-sections/
Last updated: 2026-10-30

> The conic sections calculator finds the eccentricity, linear eccentricity (focal distance), semi-latus rectum and focal parameter of a circle, ellipse, parabola or hyperbola.

## Inputs

- Conic section
- Semi-major axis a (or radius)
- Semi-minor axis b
- Focal length f (vertex to focus)

## Results

- Eccentricity e
- Linear eccentricity c (centre to focus)
- Semi-latus rectum ℓ
- Focal parameter p

## 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
```

## Explanation

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

FieldWhat to enterConic sectionChoose one: Circle, Ellipse, Parabola, HyperbolaSemi-major axis a (or radius)A numberSemi-minor axis bA numberFocal 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 e3.750.650.86.250.87737.50.9165100.9539

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Conic section](https://mathworld.wolfram.com/ConicSection.html)