Arc Length Calculator

Arc Length Formula:

\[ L = r \times \theta \]

For a circle sector:

\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx \]

Arc Length (L):

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1. What is Arc Length?

Arc length is the distance along the curved line making up the arc. For a circle, it's a portion of the circumference. The general formula for arc length can be extended to any smooth curve.

2. How Does the Calculator Work?

The calculator uses the arc length formula:

\[ L = r \times \theta \]

Where:

\( L \) — Arc length (same units as radius)
\( r \) — Radius of the circle
\( \theta \) — Central angle in radians

For general curves:

\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx \]

3. Importance of Arc Length Calculation

Applications: Arc length calculations are essential in geometry, physics, engineering, and computer graphics for designing curved structures, analyzing motion along curves, and rendering smooth shapes.

4. Using the Calculator

Tips: Enter radius in any length units, angle in radians (π radians = 180°). The result will be in the same units as the radius.

5. Frequently Asked Questions (FAQ)

Q1: How do I convert degrees to radians?
A: Multiply degrees by π/180. For example, 180° = π radians.

Q2: Does this work for any curve?
A: The simple formula works for circular arcs. For other curves, you need the integral form.

Q3: What if my angle is more than 2π?
A: The calculator still works - it will compute the length for multiple full rotations plus the remainder.

Q4: How accurate is this calculation?
A: For perfect circles, it's mathematically exact. For real-world applications, accuracy depends on your input measurements.

Q5: Can I calculate radius from arc length?
A: Yes, rearrange the formula: r = L/θ.