## Radius of Curvature Formula

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 amount by which a curve derivates itself from being flat to a curve and from a curve back to a line is called the curvature. It is a scalar quantity. The radius of curvature is the reciprocal of the curvature. 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 the Curvature Formula?

The distance from the vertex to the center of curvature is known as the radius of curvature (represented by R). 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. For any curve with equation y = f(x), with x as its parameter the radius of curvature can be given as:

## Solved Examples Using Radius of Curvature Formula

**Example 1: Find the radius of curvature of for 3x ^{2 }+ 2x – 5 at x = 1**

**Solution: **

To find: The radius of curvature.

y = 3x^{2} +2x-5

**Answer: Radius of curvature, R = 87.34 units**

**Example 2: Find the radius of curvature of for 3x ^{3} + 2x – 5 at x = 2.**

**Solution:**

To find: the radius of curvature of the given curve

y = 3x^{3} +2x-5

**Answer: Thus the radius of curvature of the curve r = e ^{θ} is √2 r**

## FAQs on Radius of Curvature Formula

### What is The Radius of the Curvature Formula?

The radius of curvature of a curve y= f(x) at a point is

curve at a point. R = 1/K, where K is the curvature of the curve and R = radius of curvature of the curve.

### How Do You Determine the Radius of Curvature?

We find the curvature of the curve at a point and take the reciprocal of it. If y = f(x), then the curve is r(t) = (t, f(t), 0) where x'(t) = 1 and x”(t) = 0, which gives the curvature as K =

### How Do You Measure Curvature of a path?

The curvature(K) of a path is measured using the radius of the curvature of the path at the given point. If y = f(x) is a curve at a particular point, then the formula for curvature is given as K = 1/R. Where R = the radius calculated using the radius of

### How is The Curvature measured?

The Curvature tells how fast the direction is changing as a point moves along a curve. The curvature is measured in radians/meters or radians/miles or degrees/mile. The curvature is the reciprocal of the radius of curvature of the curve at a given point. The radius of curvature formula is

**Application of Radius of Curvature**

In differential geometry, it is used in Cesàro equation which tells that a plain curve is an equation that relates the curvature (K) at a point of the curve to the arc length (s) from the start of the curve to a given point. Also, it is an equation relating to the radius of curvature (R) to the arc length.

Also, it can help to find the radius of curvature of the earth along a course at an azimuth.

Besides, the radius of curvature also uses three parts equation for bending of beams.

Moreover, it has a specific meaning and a sign convention in optical design. Also, spherical lenses have a center of curvature.

**Difference Between Radius and Radius of Curvature**

Radius refers to the distance between the center of a circle or any other point on the circumference of the circle and surface of the sphere. While on the other hand, the radius of curvature is the radius of the circle that touches the curve at a given point. Also, it has the same tangent and curvature at that point.

Moreover, the radius is of a real figure or shape whereas the radius of curvature is an imaginary circle.

**Types of Curvature**

In general, there are two types of curvature namely extrinsic curvature and intrinsic curvature. In this topic, we will discuss these two types.

**1. Extrinsic Curvature**

It a curvature that is a submanifold of a manifold that depends on its particular inserting. Besides, its examples include the torsion of curves in three-space and the curvature. Also, it includes the mean curvature of surfaces in three-space.

**2. Intrinsic Curvature**

It is also a curvature such as Gaussian curvature that is detectable to 2-D (two dimensional) the “populations” of a surface and not just outside observers. Also, they cannot study 3-D (three dimensional).

**Solved Question for You**

**Question. What is the curvature of the earth?**

- 7.98 inch
- 16.68 inch
- 42.6 inch
- 0.21 inch

**Answer.** The correct answer is option A because the curvature of the earth is 7.98 inch.

To calculate the curvature, it is convenient to pass from the canonical equation of the ellipse to the equation in parametric form:

Substituting the above derivatives, we get:

Solution.

Write the derivatives of the quadratic function:

**Sample Problems**

**Problem 1. Find the radius of curvature for f(x) = 4x ^{2} + 3x – 7 at x = 4.**

**Solution:**

We have, y = 4x^{2} + 3x – 7 and x = 4.

dy/dx = 16x + 3

d^{2}y/dx^{2} = 16

Using the radius of curvature formula, we get

R = (1 + (16x + 3)^{2})^{3/2}/16

= (1 + 256x^{2} + 9 + 36x)^{3/2}/16

= (256x^{2} + 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) = 3x ^{2} + 3x – 2 at x = 1.**

**Solution:**

We have, y = 3x^{2} + 3x – 2 and x = 1.

dy/dx = 6x + 3

d^{2}y/dx^{2} = 6

Using the radius of curvature formula, we get

R = (1 + (6x + 3)^{2})^{3/2}/6

= (1 + 36x^{2} + 9 + 36x)^{3/2}/6

= (36x^{2} + 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) = 3x ^{3} – 2x + 7 at x = 2.**

**Solution:**

We have, y = 3x

^{3}– 2x – 2 and x = 2.dy/dx = 9x

^{2}– 2d

^{2}y/dx^{2}= 18xUsing the radius of curvature formula, we get

R = (1 + (9x

^{2}– 2)^{2})^{3/2}/18x= (1 + 81x

^{4}+ 4 – 36x^{2})^{3/2}/18x= (81x

^{4}– 36x^{2}+ 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) = 5x ^{3} – 3x^{2} + x at x = 1.**

**Solution:**

We have, y = 5x

^{3}– 3x^{2}+ x and x = 1.dy/dx = 15x

^{2}– 6x + 1d

^{2}y/dx^{2}= 30x – 6Using the radius of curvature formula, we get

R = (1 + (15x

^{2}– 6)^{2})^{3/2}/(30x – 6)= (1 + 225x

^{4}+ 36 – 180x^{2})^{3/2}/(30x – 6)= (225x

^{4}– 180x^{2}+ 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) = x ^{2}.**

**Solution:**

We have the curve, y = x

^{2}dy/dx = 2x

d

^{2}y/dx^{2}= 2Using the radius of curvature formula, we get

R = (1 + (2x)^{2})^{3/2}/2

= (1 + 4x^{2})^{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

d

^{2}y/dx^{2}= – sin xUsing the radius of curvature formula, we get

R = – (1 + (cos x)

^{2})^{3/2}/sin x= (1 + cos

^{2}x)^{3/2}/sin x= (1 + cos

^{2}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}d

^{2}y/dx^{2}= 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 +y

^{2})^{3/2}/y

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