Arc Length Calculator

Arc Length Calculator

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Understanding Arc Length

The arc length is the distance along the curved line that forms the part of a circle's circumference. Imagine cutting a slice of pizza; the crust of that slice represents an arc, and its length is the arc length. It's a fundamental concept in geometry and trigonometry, used in various fields from engineering to architecture.

What is an Arc?

An arc is a continuous portion of the circumference of a circle. It is defined by two endpoints on the circle and the central angle that subtends it. The central angle is the angle formed by two radii connecting the center of the circle to the endpoints of the arc.

Arc Length Formulas

The formula for calculating arc length depends on whether the central angle is measured in degrees or radians.

When the Central Angle is in Radians:

The simplest formula for arc length (s) is when the central angle (θ) is expressed in radians:

s = r * θ

Where:

• s is the arc length
• r is the radius of the circle
• θ is the central angle in radians

When the Central Angle is in Degrees:

If the central angle (θ) is given in degrees, you first need to convert it to radians or use the following formula:

s = (θ / 360) * 2 * π * r

This formula essentially calculates the fraction of the total circumference (2πr) that the arc represents. A simpler form, after converting degrees to radians (θ_radians = θ_degrees * π / 180), is still s = r * θ_radians.

Where:

• s is the arc length
• r is the radius of the circle
• θ is the central angle in degrees
• π (Pi) is approximately 3.14159

How to Use the Arc Length Calculator

Our Arc Length Calculator simplifies this process for you. Just follow these steps:

1. Enter the Radius: Input the radius of the circle in the designated field. The unit of the radius will be the unit of your final arc length.
2. Enter the Central Angle: Input the central angle that subtends the arc.
3. Select Angle Unit: Choose whether your central angle is in "Degrees" or "Radians" using the dropdown menu.
4. Click "Calculate Arc Length": The calculator will instantly display the arc length.

Example Calculation

Let's walk through an example to illustrate the calculation:

Scenario: You have a circular garden bed with a radius of 8 meters. You want to install a decorative border along a section of the bed that spans a central angle of 120 degrees.

• Radius (r): 8 meters
• Central Angle (θ): 120 degrees

Using the formula for degrees:

s = (120 / 360) * 2 * π * 8

s = (1/3) * 16 * π

s = 16π / 3

s ≈ 16.7552 meters

If you were to use the calculator, you would input '8' for Radius, '120' for Central Angle, select 'Degrees', and the result would be approximately 16.7552 meters.

Applications of Arc Length

Arc length calculations are crucial in many real-world scenarios:

• Engineering: Designing curved roads, bridges, or machine parts.
• Architecture: Planning curved structures, domes, or decorative elements.
• Navigation: Calculating distances along curved paths on maps or globes.
• Physics: Analyzing circular motion or pendulum swings.
• Manufacturing: Determining the length of material needed for curved components.

This calculator provides a quick and accurate way to find the arc length for any given radius and central angle, making these calculations straightforward.