Arclength Calculator

Arclength Calculator

Use this calculator to determine the length of an arc of a circle, given its radius and the central angle in degrees.

units

degrees

function calculateArclength() { var radius = parseFloat(document.getElementById("radius").value); var angleDegrees = parseFloat(document.getElementById("angleDegrees").value); var resultDiv = document.getElementById("arclengthResult"); if (isNaN(radius) || isNaN(angleDegrees) || radius <= 0) { resultDiv.innerHTML = "Please enter valid positive numbers for Radius and Angle."; return; } // Convert angle from degrees to radians var angleRadians = angleDegrees * (Math.PI / 180); // Calculate arclength var arclength = radius * angleRadians; resultDiv.innerHTML = "The arclength is: " + arclength.toFixed(4) + " units."; } .arclength-calculator-container { font-family: 'Arial', sans-serif; background-color: #f9f9f9; padding: 20px; border-radius: 8px; box-shadow: 0 2px 5px rgba(0,0,0,0.1); max-width: 600px; margin: 20px auto; border: 1px solid #ddd; } .arclength-calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .arclength-calculator-container p { color: #555; line-height: 1.6; margin-bottom: 15px; } .calculator-input-group { margin-bottom: 15px; display: flex; align-items: center; gap: 10px; } .calculator-input-group label { flex: 1; font-weight: bold; color: #444; } .calculator-input-group input[type="number"] { flex: 2; padding: 10px; border: 1px solid #ccc; border-radius: 4px; width: calc(100% – 120px); /* Adjust width considering label and span */ } .calculator-input-group span { flex: 0 0 auto; color: #666; font-size: 0.9em; } .arclength-calculator-container button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; display: block; margin-top: 20px; transition: background-color 0.3s ease; } .arclength-calculator-container button:hover { background-color: #0056b3; } .calculator-result { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 4px; text-align: center; font-size: 1.1em; color: #155724; font-weight: bold; } .calculator-result p { margin: 0; }

Understanding Arclength

In geometry, the arclength is the distance along the curved line segment of a circle or any other curve. For a circle, an arc is a portion of its circumference. Calculating arclength is a fundamental concept in various fields, from engineering and physics to computer graphics and navigation.

The Formula for Circular Arclength

For a sector of a circle, the arclength (often denoted as 's') can be calculated using a simple formula that relates the radius of the circle and the central angle subtended by the arc. The formula is:

s = r * θ

Where:

• s is the arclength.
• r is the radius of the circle.
• θ (theta) is the central angle in radians.

Since angles are often given in degrees, it's crucial to convert degrees to radians before applying the formula. The conversion factor is:

θ (radians) = θ (degrees) * (π / 180)

Combining these, if your angle is in degrees, the formula becomes:

s = r * θ (degrees) * (π / 180)

How to Use the Arclength Calculator

1. Enter the Radius: Input the radius of the circle in the "Radius (r)" field. The unit of the radius will be the same unit for the resulting arclength (e.g., if radius is in meters, arclength will be in meters).
2. Enter the Central Angle: Input the central angle subtended by the arc in degrees in the "Central Angle (θ)" field.
3. Click "Calculate Arclength": The calculator will instantly display the arclength based on your inputs.

Examples of Arclength Calculation

• Example 1: A circle has a radius of 10 units, and the central angle is 90 degrees.
  • Angle in radians = 90 * (π / 180) = π/2 radians
  • Arclength = 10 * (π/2) = 5π ≈ 15.708 units
  (Using the calculator: Radius=10, Angle=90 -> Arclength ≈ 15.7080 units)
• Example 2: A circle has a radius of 5 meters, and the central angle is 180 degrees.
  • Angle in radians = 180 * (π / 180) = π radians
  • Arclength = 5 * π ≈ 15.708 meters
  (Using the calculator: Radius=5, Angle=180 -> Arclength ≈ 15.7080 units)
• Example 3: A circle has a radius of 25 centimeters, and the central angle is 45 degrees.
  • Angle in radians = 45 * (π / 180) = π/4 radians
  • Arclength = 25 * (π/4) ≈ 19.635 centimeters
  (Using the calculator: Radius=25, Angle=45 -> Arclength ≈ 19.6350 units)

Applications of Arclength

Arclength calculations are vital in many practical scenarios:

• Engineering: Designing curved structures, roads, or machine parts.
• Physics: Calculating the distance traveled by an object moving along a circular path.
• Cartography and Navigation: Determining distances along curved paths on maps or the Earth's surface (though for large distances, more complex spherical geometry is used).
• Computer Graphics: Rendering curved lines and shapes accurately.
• Architecture: Planning the dimensions of curved elements in buildings.

This calculator provides a quick and accurate way to find the arclength for circular sectors, simplifying complex geometric problems.