# What is the curvature of y = f(x)?

Finds the curvature κ and the radius of curvature of the graph y = f(x) at x = a, checked numerically.

- Page: https://www.acalculator.org/math/curvature-calculator
- JSON spec: https://www.acalculator.org/math/curvature-calculator.json
- Version: f7764d7c32ee

## Default answer

Example with the default inputs (Curve y = f(x) x^3 - 3x + 1, At x = a 1): The curvature of y = x^3 - 3x + 1 at x = 1 is 6.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Curve y = f(x) | The graph, typed like x^3 - 3x + 1, sin(x) or e^x. |
| a | At x = a | Where to measure the curvature: a number or a constant such as pi/2. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| curvature | Curvature κ | How sharply the graph bends at x = a, in radians per unit of length. |
| exact | Exact κ | The curvature at a, written exactly. |
| radius | Radius of curvature | 1/κ: the radius of the circle that fits the curve best at a. |
| formula | κ(x) = | \|f″(x)\| / (1 + f′(x)²)^(3/2) at every x. |

## Method

κ = |f″(x)| / (1 + f′(x)²)^(3/2), radius R = 1/κ. A computer algebra system finds f′ and f″, each checked against a difference quotient.

## Assumptions

- The variable is x; angles are in radians; ln is the natural logarithm.
- An answer that fails its check is not shown.

## Worked examples

1. f = x^3 - 3x + 1, a = 1 gives curvature = 6, exact = 6, radius = 0.166667. Source: OpenStax, Calculus Volume 3, section 3.3 Arc Length and Curvature, Example 3.13. https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature.
2. f = x - x^2/4, a = 2 gives curvature = 0.5, exact = 1/2, radius = 2.

## FAQ

### What is curvature?

Curvature measures how sharply a curve bends at a point: how fast the direction of the curve turns per unit of length along it. A straight line has curvature 0; a circle of radius r has curvature 1/r everywhere, so a small circle bends more sharply than a large one.

### What is the formula for the curvature of y = f(x)?

κ = |f″(x)| / (1 + f′(x)²)^(3/2). The second derivative measures how fast the slope changes, and the denominator converts that from change per unit of x into change per unit of length along the curve.

### What is the radius of curvature?

R = 1/κ, the radius of the osculating circle: the circle that touches the curve at the point with the same slope and the same curvature. Where κ = 0 (a straight stretch or an inflection point) the radius is infinite and the page shows none.

### Why is the curvature of y = x² largest at the vertex?

f″ = 2 everywhere, but f′ = 2x grows away from 0, so the denominator (1 + 4x²)^(3/2) grows and κ falls. At the vertex κ = 2; at x = 3 it is 2/37^(3/2), about 0.0089. The parabola is most sharply bent at its tip.

### Can I find the curvature of a parametric curve or a helix?

Not on this page. It handles graphs y = f(x). For a curve r(t) = (x(t), y(t)) the formula is |x′y″ − y′x″|/(x′² + y′²)^(3/2), and for curves in space κ = ‖r′ × r″‖/‖r′‖³.

### How is the answer checked?

A computer algebra system finds f′ and f″; each is compared with a numeric difference quotient at 20 points. The exact value at a is simplified by the algebra and compared with the formula at a. If a check fails, the page shows the decimal only or says "No verified answer".

## Sources

- OpenStax, Calculus Volume 3, section 3.3 Arc Length and Curvature: https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature