The Radius of Curvature Formula

The Radius of Curvature Formula

Any approximate radius of a circle at any point is called the radius of curvature. 

As we move further to the curve, the radius of the curvature alters or modifies. The radius of curvature is denoted by ”R”.

Curvature is the magnitude by which a curved shape derives itself from existing as flat to a curve and from a bender back to a line. It is a scalar quantity. The radius of curvature is the interchange of the curvature. The radius of curvature is not an actual shape or illustration; instead, it is an imagined circle. 

The Formula for the Radius of Curvature 

The spatial arrangement from the vertex to the middle of curvature is known as the radius of curvature (represented as R). Any circles’ radius approximate radius at any point is called the radius of curvature of that curve, or the vector length of curvature.

For any given curve, having equation as

y = f(x), 

Where,

x is the parameter, then the radius of the curvature can be given as:

R = (1+(dy / dx)²)^3/2 / |d²y / dx²|

In polar coordinates r=r(Θ), the radius of curvature formula is yielded as:

ρ=1 / K[r²+(dr / dθ)²]3/2 / r2+2(dr / dθ)²−rd²r / dθ²

R= 1/K, where R is the length or radius of curvature and K is the derivative of curvature.

Solved Examples

1. Find the radius of curvature of for 3x² + 2x – 5 at x = 1

Answer:

The radius of curvature.

y = 3x² +2x-5

dy / dx = 6x + 2

d²y / dx² = 6

By using the formula of curvature, we get:

R = (1+(dy / dx)²)3/2 / |d2y / dx²|

By putting down the values, we get,

R=(1+(6x+2)²)3/2 / 6

R=(1+(36x+4 + 24x)3/2 / 6

Now putting x = 1

R=(36+5 + 24)3/2 / 6

R = (65)3/2 / 6

R = 87.34

Hence, the radius of curvature is 87.34.

2. Observe the radius of curvature of 3x³ + 2x – 5 at x = 2.

Answer:

y = 3x³ +2x-5

dy / dx = 9x² + 2

d2y / dx² = 18x 

By using the formula of radius of curvature, we get,

R = (1+(dy / dx)²)3/2 / |d2y / dx²|

Inserting the values we get,

R=(1+(81×4+4 + 36x²)3/2 / 18x

Putting x = 2

We get,

R = (1296 + 5 + 144)3 / 2

R = 1413.19

Hence, the radius of curvature is 1413.19 units.