Radius of Curvature — Definition, Formula & Examples

Radius of curvature is the radius of the circle that best approximates a curve at a given point. A small radius means the curve bends sharply; a large radius means it bends gently.

For a smooth curve at a point where the curvature $\kappa \neq 0$ , the radius of curvature $R$ is defined as $R = 1/\kappa$ , where $\kappa$ is the curvature. Geometrically, $R$ is the radius of the osculating circle — the unique circle that matches the curve's position, tangent direction, and rate of turning at that point.

Key Formula

R = \frac{\left(1 + \left(f'(x)\right)^2\right)^{3/2}}{\left|f''(x)\right|}

Where:

$R$ = Radius of curvature at the point
$f'(x)$ = First derivative of the function at the point
$f''(x)$ = Second derivative of the function at the point

How It Works

To find the radius of curvature of

y = f(x)

, first compute

f'(x)

and

f''(x)

. Plug these into the formula to get

R

at any point where

f''(x) \neq 0

. The center of the osculating circle lies along the unit normal to the curve at distance

R

. When

f''(x) = 0

, the curvature is zero and the radius of curvature is infinite, meaning the curve is locally straight.

Worked Example

Problem: Find the radius of curvature of

y = x^2

at the point

(1, 1)

Find the derivatives: Differentiate

y = x^2

twice.

f'(x) = 2x, \quad f''(x) = 2

Evaluate at $x = 1$: Substitute

x = 1

into both derivatives.

f'(1) = 2, \quad f''(1) = 2

Apply the formula: Plug the values into the radius of curvature formula.

R = \frac{(1 + 2^2)^{3/2}}{|2|} = \frac{(1+4)^{3/2}}{2} = \frac{5^{3/2}}{2} = \frac{5\sqrt{5}}{2} \approx 5.59

Answer: The radius of curvature at

(1,1)

\dfrac{5\sqrt{5}}{2} \approx 5.59

Why It Matters

Engineers use the radius of curvature to design roads and railways — vehicles skid when a turn's radius is too small for the speed. In optics, the radius of curvature of a lens or mirror surface directly determines its focal length. It also appears in beam bending analysis in structural engineering, where it relates bending moment to deflection.

Common Mistakes

Mistake: Forgetting the absolute value on

f''(x)

, which can produce a negative radius.

Correction: Radius of curvature is always positive. The absolute value of the second derivative ensures

R > 0

regardless of whether the curve is concave up or concave down.