Hyperbola Calculator
Analyze a hyperbola in standard form: center, vertices, co-vertices, foci, eccentricity, and asymptotes, with a plotted diagram of both branches and step-by-step math.
Input
Transverse semi-axis — distance from the center to each vertex. Must be positive.
Conjugate semi-axis, sets the asymptote slope. Must be positive.
Output
The hyperbola will be plotted here.
| Property | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/hyperbola-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"orientation": "horizontal",
"a": "3",
"b": "2",
"h": "0",
"k": "0"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `hyperbola-calculator` tool (Hyperbola Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/hyperbola-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Hyperbola Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
The Hyperbola Calculator analyzes a hyperbola in standard form, either horizontal (opens left/right) or vertical (opens up/down), centered anywhere in the plane. It returns the center, vertices, co-vertices, foci, eccentricity, and both asymptote equations, with a plotted diagram of both branches and a full step-by-step derivation.
It's built for conic-sections and precalculus students checking a homework answer, and for anyone who needs a hyperbola's key points without deriving c = √(a² + b²) and the asymptote slopes by hand.
How to use it
- Choose the orientation — horizontal for
(x − h)²/a² − (y − k)²/b² = 1, vertical for(y − k)²/a² − (x − h)²/b² = 1. - Enter a (the transverse semi-axis — distance from the center to each vertex) and b (the conjugate semi-axis, which sets the asymptote slope). Both must be positive.
- Enter the center (h, k) — leave both at
0for a hyperbola centered at the origin. - Read the Diagram for a plot of both branches with the center, vertices, foci, and asymptotes marked, the Analysis table for every computed property, and Step-by-Step Solution for the full derivation.
Results update automatically as you change any field. For example, a horizontal hyperbola with a = 3, b = 2, centered at the origin has vertices at (3, 0) and (-3, 0), foci at (±3.60555, 0), and asymptotes y = 0.666667x and y = -0.666667x.
What do a, b, and c mean?
- a is the transverse semi-axis: half the distance between the two vertices, measured along the axis the hyperbola opens along.
- b is the conjugate semi-axis: it doesn't lie on the curve itself, but together with a it defines the "auxiliary rectangle" whose diagonals are the asymptotes.
- c is the distance from the center to each focus, given by
c = √(a² + b²)— note this is a plus, unlike an ellipse'sc = √(a² − b²), which is exactly why a hyperbola's eccentricitye = c/ais always greater than 1.
Why do the asymptotes matter?
As you move far from the center along either branch, the hyperbola gets arbitrarily close to two straight lines through the center — the asymptotes — without ever touching them. Their slope is b/a for a horizontal hyperbola or a/b for a vertical one. Sketching those two lines first, through the auxiliary rectangle's corners, is the standard way to hand-draw an accurate hyperbola.
Working with a circle or an ellipse instead? See the Circle & Ellipse Calculator. For a parabola (y = ax² + bx + c), see the Parabola Calculator.
Privacy
This calculator runs entirely in your browser. Your inputs are never uploaded, logged, or stored — the computation happens locally on your device.