Length Arc Calculator

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Arc Length Calculator

Enter the radius of the circle (e.g., 10 cm).
Enter the central angle (e.g., 90 for degrees, 1.57 for radians).
Degrees Radians
function calculateArcLength() { var radius = parseFloat(document.getElementById('radius').value); var angleValue = parseFloat(document.getElementById('angleValue').value); var angleUnit = document.getElementById('angleUnit').value; var resultDiv = document.getElementById('arcLengthResult'); var arcLength; if (isNaN(radius) || isNaN(angleValue) || radius <= 0) { resultDiv.innerHTML = 'Please enter valid positive numbers for Radius and Angle.'; return; } if (angleUnit === 'degrees') { // Convert degrees to radians: radians = degrees * (PI / 180) var angleRadians = angleValue * (Math.PI / 180); arcLength = radius * angleRadians; } else { // radians arcLength = radius * angleValue; } resultDiv.innerHTML = 'The Arc Length is: ' + arcLength.toFixed(4) + ' units'; } // Initial calculation on page load for default values window.onload = calculateArcLength;

Understanding Arc Length

An arc is a portion of the circumference of a circle. Imagine cutting a slice out of a circular pizza; the crust of that slice represents an arc. The length of this arc is a fundamental concept in geometry, with practical applications across various fields.

What is Arc Length?

The arc length is simply the distance along the curved line that forms the part of the circle's circumference. It's determined by two main factors: the radius of the circle and the central angle that subtends (or "cuts off") that arc.

  • Radius (r): The distance from the center of the circle to any point on its circumference.
  • Central Angle (θ): The angle formed by two radii connecting the center of the circle to the endpoints of the arc. This angle can be measured in degrees or radians.

Why is Arc Length Important?

Calculating arc length is crucial in many real-world scenarios:

  • Engineering: Designing curved structures like bridges, tunnels, or machine parts.
  • Architecture: Planning curved walls, domes, or decorative elements.
  • Cartography: Measuring distances along curved paths on maps.
  • Physics: Analyzing circular motion or the path of projectiles.
  • Manufacturing: Cutting materials into specific curved shapes.

How to Calculate Arc Length

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

1. When the Central Angle is in Radians (θ):

This is the most straightforward formula:

Arc Length (s) = r × θ

Where:

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

2. When the Central Angle is in Degrees (θ):

If your angle is in degrees, you first need to convert it to radians. There are π radians in 180 degrees.

Arc Length (s) = r × (θ × π / 180)

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

Using the Arc Length Calculator

Our calculator simplifies this process for you:

  1. Enter the Radius: Input the radius of your circle in any unit (e.g., meters, inches, feet).
  2. Enter the Central Angle Value: Input the numerical value of your central angle.
  3. Select Angle Unit: Choose whether your angle is in "Degrees" or "Radians" from the dropdown menu.
  4. Click "Calculate Arc Length": The calculator will instantly display the arc length in the same unit as your radius.

Examples:

Example 1: Angle in Degrees

You have a circle with a radius of 10 cm, and you want to find the length of an arc subtended by a central angle of 90 degrees.

  • Radius (r) = 10 cm
  • Central Angle (θ) = 90 degrees

Using the formula: Arc Length = 10 × (90 × π / 180) = 10 × (π / 2) ≈ 10 × 1.5708 = 15.708 cm.

(Input 10 for Radius, 90 for Angle Value, select Degrees)

Example 2: Angle in Radians

Consider a circular path with a radius of 5 meters. You need to find the length of an arc corresponding to a central angle of 1.5 radians.

  • Radius (r) = 5 meters
  • Central Angle (θ) = 1.5 radians

Using the formula: Arc Length = 5 × 1.5 = 7.5 meters.

(Input 5 for Radius, 1.5 for Angle Value, select Radians)

This calculator is a handy tool for students, engineers, designers, and anyone needing to quickly determine arc lengths for various applications.

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