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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 κ≠0\kappa \neq 0, the radius of curvature RR is defined as R=1/κR = 1/\kappa, where κ\kappa is the curvature. Geometrically, RR 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=(1+(f′(x))2)3/2∣f′′(x)∣R = \frac{\left(1 + \left(f'(x)\right)^2\right)^{3/2}}{\left|f''(x)\right|}
Where:
  • RR = Radius of curvature at the point
  • f′(x)f'(x) = First derivative of the function at the point
  • f′′(x)f''(x) = Second derivative of the function at the point

How It Works

To find the radius of curvature of y=f(x)y = f(x), first compute f′(x)f'(x) and f′′(x)f''(x). Plug these into the formula to get RR at any point where f′′(x)≠0f''(x) \neq 0. The center of the osculating circle lies along the unit normal to the curve at distance RR. When f′′(x)=0f''(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=x2y = x^2 at the point (1,1)(1, 1).
Find the derivatives: Differentiate y=x2y = x^2 twice.
f′(x)=2x,f′′(x)=2f'(x) = 2x, \quad f''(x) = 2
Evaluate at $x = 1$: Substitute x=1x = 1 into both derivatives.
f′(1)=2,f′′(1)=2f'(1) = 2, \quad f''(1) = 2
Apply the formula: Plug the values into the radius of curvature formula.
R=(1+22)3/2∣2∣=(1+4)3/22=53/22=552≈5.59R = \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)(1,1) is 552≈5.59\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)f''(x), which can produce a negative radius.
Correction: Radius of curvature is always positive. The absolute value of the second derivative ensures R>0R > 0 regardless of whether the curve is concave up or concave down.

Related Terms