Radius of Curvature Formula
The vector length of curvature is the radius of curvature. Any approximate circle's radius at any particular given point is called the radius of curvature of the curve. As we move along the curve the radius of curvature changes. The radius of curvature formula is denoted as 'R'. The radius of curvature is not a real shape or figure rather it's an imaginary circle. Let us understand the radius of curvature formula in detail using solved examples in the following section.
What is the Radius of Curvature Formula?
Any approximate circle's radius at any particular given point is called the radius of curvature of the curve or the vector length of curvature is also called the radius of curvature and the formula for the radius of curvature can be given as:
Radius of Curvature Formula =\(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{\dfrac{d^{2}y}{dx} }\)

In polar coordinates r=r(Θ), the radius of curvature formula is given as:
\(\rho=\frac{1}{\mathrm{K}} \frac{\left[r^{2}+\left(\frac{d r}{d \theta}\right)^{2}\right]^{3 / 2}}{\left  r^{2}+2\left(\frac{d r}{d \theta}\right)^{2}r \frac{d^{2} r}{d \theta^{2}}\right }\)
Let's take a quick look at a couple of examples to understand the radius of curvature formula, better.
Solved Examples Using Radius of Curvature Formula

Example 1:Find the radius of curvature of for 3x^{2 }+2x5 at x=1
Solution:
To find: The radius of curvature.
y = 3x^{2} +2x5
\(\dfrac{dy}{dx} = 6x + 2\)
\(\dfrac{d^2y}{dx^2} = 6\) (given)Using radius of curvature formula,
\(R = \dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{\dfrac{d^{2}y}{dx}}\)
Put the values,
\(R = \dfrac{(1+(6x+2)^{2})^{3/2}}{6}\)
\(R = \dfrac{(1+36x^2+4+24x)^{3/2}}{6}\)
\(R = \dfrac{(36x^2+5+24x)^{3/2}}{6}\)
Putting x = 1
\(R = \dfrac{(36+5+24)^{3/2}}{6}\)
\(R = \dfrac{(65)^{3/2}}{6}\)
R = 87.34
Answer: Radius of curvature, R = 87.34 units

Example 2: the radius of curvature of for 3x^{3} +2x5 at x=2.
Solution:
To find: Interest rate
y = 3x^{3} +2x5
\(\dfrac{dy}{dx} = 9x^2 + 2\)
\(\dfrac{d^2y}{dx^2} = 18x\) (given)Using radius of curvature formula,
\(R = \dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{\dfrac{d^{2}y}{dx}}\)
Put the values,
\(R = \dfrac{(1+(9x^2+2)^{2})^{3/2}}{18x}\)
R = \(\dfrac{(1+81x^4+4+36x^2)^{3/2}}{18x}\)
R = \(\dfrac{(81x^4+5+36x^2)^{3/2}}{18x}\)
Putting x = 2
R = \(\dfrac{(1296+5+144)^{3/2}}{36}\)
R = \(\dfrac{(1373)^{3/2}}{36}\)
R = 1413.19
Answer: Radius of curvature, R = 1413.19 units