Tangent Line to Circle Calculator
Find the tangent line equation(s) and point(s) of tangency from a point to a circle, given the circle's center and radius. Handles a point outside the circle (two tangents), on the circle (one tangent), or inside it (none), and reports the tangent length.
Input
Output
| Metric | Value |
|---|---|
| No data yet | |
| Line | Equation | Point of tangency |
|---|---|---|
| No data yet | ||
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/tangent-line-to-circle-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"centerX": "0",
"centerY": "0",
"radius": "5",
"pointX": "13",
"pointY": "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 `tangent-line-to-circle-calculator` tool (Tangent Line to Circle 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/tangent-line-to-circle-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Tangent Line to Circle 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 |
Also available with
Guides
The Tangent Line to Circle Calculator finds every tangent line from a given point to a circle, along with each point of tangency and the tangent length. Enter the circle's center and radius plus a point, and it automatically classifies whether the point is outside the circle (two tangent lines), exactly on it (one tangent line), or inside it (no tangent line exists at all).
How to use it
- Enter the circle's center coordinates and radius.
- Enter the point you want tangent lines from.
- Read the case classification and, if any tangent lines exist, their equations and points of tangency.
The math
Let the circle have center (h, k) and radius r, and the point be (x₀, y₀). Let d be the distance from the point to the center:
d = √((x₀ − h)² + (y₀ − k)²)
- d > r (point outside the circle): two tangent lines exist. The tangent length is L = √(d² − r²). The points of tangency sit at angle θ₀ ± α around the center, where θ₀ is the direction from the center to the point and α = arccos(r / d).
- d = r (point on the circle): exactly one tangent line exists, perpendicular to the radius at that point — the point of tangency is the point itself.
- d < r (point inside the circle): no tangent line exists, since every line through the point crosses the circle.
Worked example
Circle centered at (0, 0) with radius 5, and the point (13, 0): d = 13, so the point is outside the circle. The tangent length is √(13² − 5²) = √144 = 12. The two points of tangency are (25/13, 60/13) ≈ (1.92308, 4.61538) and (25/13, −60/13) ≈ (1.92308, −4.61538), giving tangent lines y = −0.416667x + 5.41667 and y = 0.416667x − 5.41667.
Frequently asked questions
What if the point is exactly on the circle?
There's exactly one tangent line, and it's always perpendicular to the radius drawn to that point — the point of tangency is the point you entered.
What if the point is inside the circle?
No tangent line exists — every straight line through an interior point must cross the circle's boundary (or pass through it), so it can never just touch it once.
Can the radius be zero or negative?
No — a circle needs a positive radius. Enter a value greater than 0.
Is my data sent anywhere?
No. All calculations run locally in your browser. Your input never leaves your device.