How to Use Conic Section Calculator
The Conic Section Calculator lets you analyze any of the four classical conic sections by entering the parameters of its standard-form equation. Select the type, fill in the values, and the calculator instantly returns the key geometric properties.
- Select the conic type — choose Circle, Ellipse, Parabola, or Hyperbola.
- Enter the parameters — for a circle, enter center (h, k) and radius r; for an ellipse or hyperbola, enter center and semi-axes a and b; for a parabola, enter the vertex, focus distance p, and orientation.
- Review the output — the Conic Section Calculator displays the standard equation, center or vertex, foci, eccentricity, and any other relevant properties.
For the parabola, select whether it opens vertically (up/down) or horizontally (left/right) using the orientation toggle.
Formula & Theory - Conic Section Calculator
The Conic Section Calculator uses the following standard-form equations:
Circle:
(x − h)² + (y − k)² = r²
Ellipse:
(x − h)²/a² + (y − k)²/b² = 1
Parabola (vertical):
(x − h)² = 4p(y − k)
Parabola (horizontal):
(y − k)² = 4p(x − h)
Hyperbola:
(x − h)²/a² − (y − k)²/b² = 1| Symbol | Meaning |
|---|---|
| (h, k) | Center (circle, ellipse, hyperbola) or vertex (parabola) |
| r | Radius of circle |
| a, b | Semi-major and semi-minor axes (ellipse); transverse and conjugate axes (hyperbola) |
| c | Focal distance, c² = a² − b² (ellipse), c² = a² + b² (hyperbola) |
| p | Distance from vertex to focus (parabola) |
| e | Eccentricity = c/a |
The Conic Section Calculator computes eccentricity (e = c/a), foci coordinates, asymptotes for hyperbolas, and directrix for parabolas using these relationships directly.
Assumptions and Limits
All input parameters (r, a, b) must be positive. For the hyperbola, the tool assumes the transverse axis is horizontal. The parabola form assumes the standard orientation where p > 0 means the parabola opens upward or rightward.
Use Cases for Conic Section Calculator
The Conic Section Calculator is valuable for students and professionals working with analytic geometry. Common uses include:
- Algebra and precalculus courses — Identify conic sections from given equations and compute standard properties for exercises.
- Graphing and visualization — Determine foci, axes, and vertices before plotting on paper or in graphing software.
- Engineering design — Compute focal properties of parabolic reflectors, elliptical orbits, or hyperbolic navigation curves.
- Exam preparation — Quickly verify the eccentricity and focal distances for conic section problems on standardized tests.
The Conic Section Calculator shows the full standard-form equation with your values substituted, making it easy to check your work and understand how the formula is applied.