Curvature Calculator
Calculate the curvature κ and radius of curvature R of a curve y = f(x) at a point, for any function — not just a quadratic — with both derivatives shown symbolically and a step-by-step derivation.
Input
A function of x, written in standard math notation (^ for powers, sqrt(), sin(), cos(), etc.).
The point to evaluate the curvature at.
Output
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/curvature-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"functionExpr": "x^2",
"x": "1"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `curvature-calculator` tool (Curvature 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/curvature-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Curvature 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 Curvature Calculator finds the curvature (κ) and radius of curvature (R) of a curve y = f(x) at a given point x — how sharply the curve is bending there, and the radius of the circle that best hugs the curve at that exact point. It shows both derivatives symbolically, along with a full step-by-step derivation.
Unlike most curvature calculators, which only accept a quadratic y = ax² + bx + c, this one works with any function of x — polynomials, trig functions, roots, logarithms, and combinations of these.
How to use it
- Enter your function in terms of x — standard math notation works:
^for powers, plussqrt(),sin(),cos(),log(), and the other functions supported by mathjs's expression parser. - Enter x, the point to evaluate the curvature at.
- Read Curvature (κ) for the numeric curvature and Radius of Curvature (R) for its reciprocal.
- f'(x) and f''(x) show the first and second derivatives symbolically, and Step-by-Step Derivation shows the full calculation.
Results update automatically as you type. For example, f(x) = x^2 at x = 1 returns a curvature of 0.178885 and a radius of curvature of 5.59017.
What is curvature?
For a plane curve y = f(x), curvature measures how quickly the curve's direction changes as you move along it:
κ(x) = |f''(x)| / (1 + f'(x)²)^(3/2)A straight line has curvature 0 everywhere (it never turns). A tight bend has high curvature; a gentle bend has low curvature. The radius of curvature R = 1/κ is the radius of the osculating circle — the circle that best approximates the curve at that point, sharing its tangent line and rate of turning. A small radius means a sharp bend; a large radius means a gentle one. When κ = 0 (a straight line, or an inflection point where the curve momentarily stops bending), R is undefined — there's no finite circle that fits a curve with no curvature.
What functions can I enter?
Anything mathjs's expression evaluator understands: polynomials (x^3 - 2x + 1), trigonometric functions (sin(x), cos(x), tan(x)), roots and logarithms (sqrt(x), log(x)), and combinations of these. Angles in trig functions are in radians by default — write pi for π.
To evaluate arbitrary math expressions directly, see the Math Evaluator, or to find the average rate of change of f(x) over an interval, see the Average Rate of Change Calculator.
Privacy
This calculator runs entirely in your browser. Your function and point are never uploaded, logged, or stored — the computation happens locally on your device.