Our length of curve calculator helps you accurately find the length of any function-based curve.

## How to Use the Length of Curve Calculator

To calculate the length of a curve defined by a function f(x):

- Enter the mathematical expression for your function of x in the “Function of x” field.
- Specify the Start value (a) and End value (b) of x over which you want to calculate the length.
- Enter the number of intervals (n) for the approximation (higher values of n lead to more accurate results but take more time to compute).
- Click the Calculate button to get the result.

## How It Works

This calculator approximates the length of the curve using the sum of line segment lengths between successive points on the curve. For a function f(x), the length L from a to b can be approximated by:

L ≈ Σ sqrt((Δx)^2 + (Δy)^2) for i = 1 to n

where Δx = (b – a) / n and Δy = f(xi+1) – f(xi).

## Limitations

- The accuracy of the result depends on the number of intervals (n). Generally, higher values yield more accurate results but can take longer to compute.
- The function input must be valid JavaScript code that returns a number when evaluated for a given x.
- Highly oscillatory functions or functions with vertical asymptotes may result in large errors even with a high number of intervals.

## Use Cases for This Calculator

### Calculate the Length of a Straight Line

Input the coordinates of the two endpoints of the straight line, and the calculator will instantly give you the length of the line. It’s perfect for quickly determining the distance between two points on a map or coordinate plane.

### Estimate the Length of a Wavy Curve

Describe the wavy curve by providing a mathematical function or data points, and our calculator will calculate the approximate length of the curve. This is handy for measuring the length of rivers, roads, or any curved path.

### Determine Length of a Circular Arc

Enter the radius and angle of the circular arc, and the calculator will determine the length of the arc. This feature is useful for calculating the distance along a circular track or the circumference of a sector.

### Find Length of a Parabolic Curve

Specify the equation of the parabolic curve, and our calculator will find the length of the curve within a given interval. This is ideal for determining the length of arches or bridges that follow a parabolic shape.

### Compute Length of an Elliptical Curve

Input the semi-major and semi-minor axes of the ellipse, and our calculator will compute the length of the ellipse. This can be helpful for calculating the circumference of oval-shaped objects or orbits.

### Calculate Length of a Bezier Curve

Provide the control points of the Bezier curve, and the calculator will compute its length. This feature is convenient for graphic designers or anyone working with curved lines in design software.

### Determine Length of a Spline Curve

Input the data points that define the spline curve, and our calculator will find its length. This is useful for calculating the total length of irregularly curved paths or boundaries.

### Estimate Length of a Helical Curve

Specify the pitch, radius, and total angle of the helix, and the calculator will estimate the length of the helical curve. This is handy for calculating the length of screws, springs, or DNA strands.

### Measure Length of a Spiral Curve

Describe the spiral curve by providing its equation or parameters, and our calculator will measure the length of the spiral. This feature is great for determining the total length of spiral staircases or coils.

### Calculate Length of a Polygonal Path

Input the coordinates of the vertices of a polygonal path, and the calculator will determine the total length of the path. This is perfect for measuring the perimeter of irregularly shaped regions or routes.