Radius of Curvature Formula

A radius of curvature is an important idea in numerical calculations and physics especially when dealing with curves of all kinds. It describes how sharply a curve bends at any one point. The concept of the radius of curvature can be utilized in different areas such as road and track construction/designing processes, studying optical systems analysis, or observing biological structures among others.

In this article therefore we shall look into the formula for calculating radius of curvature, provide exercises to improve your understanding as well as respond to some frequently asked questions about it.

What is The Radius of Curvature?

Any approximate radius of a circle at any point is called the radius of curvature. The curvature is that scalar value by which a curve deviates from being flat to a curve and from a curve back to a line. The reciprocal of the curvature is the radius of curvature and it is an imagined circle rather than a genuine form or figure. The radius of curvature is a measure of how sharply a curve bends at a given point.

The radius of curvature is the radius of the approximate circle at a specific place. It is represented by the curvature vector length and denoted by the symbol R. It is defined as the distance between the vertex and the center of curvature. As the curve advances, the radius changes. The radius of curvature measures how sharply the curve bends at that point; a smaller radius indicates a tighter bend, while a larger radius indicates a gentler curve. The key idea is that the radius of curvature provides insight into the local geometric properties of the curve.

Radius of Curvature Formula

The radius of curvature of a curve is defined as the approximate radius of a circle at any given point or the vector length of a curvature. It exists for any curve with the equation y = f(x) with x as its parameter.

Radius of curvature, R = \frac{\left(1+(\frac{dy}{dx})^2\right)^\frac{3}{2}}{\left|\frac{d^2y}{dx^2}\right|}

where,

dy/dx = first derivative of the function y = f(x),

d²y/dx² = second derivative of the function y = f(x).

In case of polar coordinates r=r(θ), the radius of curvature is given by

\rho=\frac{1}{K}\frac{\left[r^2+(\frac{dr}{d\theta})^2\right]^{\frac{3}{2}}}{\left|r^2+2(\frac{dr}{d\theta})^2-r\frac{d^2r}{d\theta^2}\right|}

where,

K is the tangent vector function and curvature of the curve given by dT/ds,

r is the radius of curvature.

Sample Problems on Radius of Curvature Formula

Problem 1. Find the radius of curvature for f(x) = 4x² + 3x - 7 at x = 4.

Solution:

We have, y = 4x² + 3x - 7 and x = 4.

dy/dx = 16x + 3

d²y/dx² = 16

Using the radius of curvature formula, we get

R = (1 + (16x + 3)²)^3/2/16

= (1 + 256x² + 9 + 36x)^3/2/16

= (256x² + 36x + 10)^3/2/16

Substitute the value x = 4.

R = (256 (4)² + 36 (4) + 10)^3/2/16

= (4096 + 144 + 10)^3/2/16 

= 27066/16

= 1691.625 units

Problem 2. Find the radius of curvature for f(x) = 3x² + 3x - 2 at x = 1.

Solution:

We have, y = 3x² + 3x - 2 and x = 1.

dy/dx = 6x + 3

d²y/dx² = 6

Using the radius of curvature formula, we get

R = (1 + (6x + 3)²)^3/2/6

= (1 + 36x² + 9 + 36x)^3/2/6

= (36x² + 36x + 10)^3/2/6

Substitute the value x = 1.

R = (36 (1)² + 36 (1) + 10)^3/2/6

= (36 + 36 + 10)^3/2/6

= 742.54/6

= 123.75 units

Problem 3. Find the radius of curvature for f(x) = 3x³ - 2x + 7 at x = 2.

Solution:

We have, y = 3x³ - 2x - 2 and x = 2.

dy/dx = 9x² - 2

d²y/dx² = 18x

Using the radius of curvature formula, we get

R = (1 + (9x² - 2)²)^3/2/18x

= (1 + 81x⁴ + 4 - 36x²)^3/2/18x

= (81x⁴ - 36x² + 5)^3/2/18x

Substitute the value x = 2.

R = (81 (2)⁴ - 36 (2) + 5)^3/2/18 (2)

= (576 - 72 + 5)^3/2/36

= 12305.26/36

= 512.71 units

Problem 4. Find the radius of curvature for f(x) = 5x³ - 3x² + x at x = 1.

Solution:

We have, y = 5x³ - 3x² + x and x = 1.

dy/dx = 15x² - 6x + 1

d²y/dx² = 30x - 6

Using the radius of curvature formula, we get

R = (1 + (15x² - 6)²)^3/2/(30x - 6)

= (1 + 225x⁴ + 36 - 180x²)^3/2/(30x - 6)

= (225x⁴ - 180x² + 37)^3/2/(30x - 6)

Substitute the value x = 1.

R = (225(1)⁴ - 180(1)² + 37)^3/2/(30 (1) - 6)

= (225 - 180 + 37)^3/2/24

= 742.54/24

= 30.93 units

Problem 5. Find the radius of curvature for the curve f(x) = x².

Solution:

We have the curve, y = x²

dy/dx = 2x

d²y/dx² = 2

Using the radius of curvature formula, we get

R = (1 + (2x)²)^3/2/2

= (1 + 4x²)^3/2/2

= (1 + 4y)^3/2/2

Problem 6. Find the radius of curvature for the curve f(x) = sin x.

Solution:

We have the curve, y = sin x.

dy/dx = cos x

d²y/dx² = - sin x

Using the radius of curvature formula, we get

R = - (1 + (cos x)²)^3/2/sin x

= (1 + cos² x)^3/2/sin x

= (1 + cos² x)^3/2/y

Problem 7. Find the radius of curvature for the curve f(x) = e^x.

Solution:

We have the curve, y = e^x.

dy/dx = e^x

d²y/dx² = e^x

Using the radius of curvature formula, we get

R = - (1 + (e^x)²)^3/2/e^x

= (1 + e^2x)^3/2/e^x

= (1 +y²)^3/2/y

Practice Problems on Radius of Curvature Formula

1. Find the radius of curvature of the curve y=x^2 at x=1.

2. Determine the radius of curvature for the parametric curve x=\cos(t) and y=\sin(t) at t=\frac{\pi}{4}.

3. Calculate the radius of curvature of the circle x^2 + y^2=25 at any point on the circle.

4. For the curve given by r=2+3\sin(\theta), find the radius of curvature at \theta=\frac{\pi}{6}.

5. Find the radius of curvature of the ellipse \frac{x^2}{4}+\frac{y^2}{9}=1 at the point (2,0).

6. Compute the radius of curvature for the curve y=e^x at x=0.

7. Determine the radius of curvature of the curve x=t^2 and y=t^3 at t=1.

8. Find the radius of curvature of the hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 at the point (a,0).

9. For the logarithmic spiral defined by r=e^{\theta}, find the radius of curvature at \theta=1.

10. Calculate the radius of curvature of the curve y=\sin{(x)} at x=\frac{\pi}{2}.