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# Length of Polar Curve Calculator + Online Solver With Free Steps

The** Length of Polar Curve Calculator **is an online tool to find the arc length of the polar curves in the Polar Coordinate system.

A **polar curve** is a shape obtained by joining a set of polar points with different distances and angles from the origin. This set of the polar points is defined by the **polar function**.

The result displays the exact value of **length** and **polar plot **for the input function.

## What Is a Length of Polar Curve Calculator?

**A Length of Polar Curve Calculator is an online calculator that can be used to determine the arc length of polar function over a specified interval.**

The **arc** **length** is a measure of distance between two points along a segment of the polar curve. This simple **calculator** computes the arc length by quickly solving the standard integration formula defined for evaluating the arc length.

The **formula** for arc length of polar curve is shown below:

\[ Length = \int_{\theta=a}^{b} \sqrt{r^2 + (\dfrac{dr}{d\theta})^2} d\theta \]

Where the **radius** equation (r) is a function of the **angle** ($\theta$). The integral limits are the upper and lower limit of angle. The function is differentiated concerning the angle which is denoted by $dr/d\theta$.

Therefore, finding out the length needs several **steps **to be done, which is a time-intensive procedure and there is a chance of mistakes if solved by hand. But you can save your precious time by using this **superb** tool that provides you with the most **accurate** results.

This online **calculator** is readily available in your browser at any time and place. You don’t need any prior knowledge or require any skill to operate this calculator.

## How To Use the Length of Polar Curve Calculator?

You can use the **Length of Polar Curve Calculator** by inserting the values of the input components in their mentioned fields. Follow the given steps to get good results.

### Step 1

Enter the polar equation which is a function of angle ($\theta$) in the **Polar Equation R **tab. It can be any algebraic or trigonometric equation.

### Step 2

Enter the starting point of the angle in the box named **From** and the endpoint in the **To **box. The points can be any value between 0 and $2\pi$.

### Step 3

Press the **Submit** button to get the desired result.

### Result

The final result is provided in two steps. The first part is the **length of the polar curve** between the points you specified and the second part is the **polar graph** that is drawn within that particular span.

The polar graph displays the total polar curve in the **dotted lines,** whereas the specific portion of the curve for which arc length is evaluated is shown in a **straight line**.

## Solved Examples

To further clarify the use of the calculator, let’s explore some solved examples from this handy calculator.

### Example 1

Consider the following polar equation:

**r($\theta$) = 6sin($\theta$) **

The interval of angle for calculating the arc length is given as:

\[ \theta = (0,\pi/2) \]

### Solution

The calculator gives the following results.

**Length of Polar Curve:**

\[ \int_{0}^{\pi/2} 6 d\theta = 3\pi \approx 9.4248 \]

**Polar Plot:**

The polar plot is depicted in Figure 1. The **straight bold **line represents the section of the curve for which arc length is calculated while the **dotted **line shows the remaining portion of the curve.

### Example 2

Consider the below-mentioned radius equation:

**r($\theta$) = 5+cos(4$\theta$) **

The integral limits for angle are as follows:

**$\theta$ = (0,$\pi$) **

### Solution

For the above polar function, our calculator attains the following arc length and polar plot.

**Length of Polar Curve:**

\[ \int_{0}^{\pi} \sqrt{ (5+\cos(4\theta))^2 + \sin^{2} (4\theta) } d\theta \approx 17.9971 \]

**Polar Plot:**

The polar plot is shown in Figure 2 below:

*All the Mathematical Images/Graphs are created using GeoGebra.*