How to Calculate Length of an Arc

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Arc Length Calculator

Use this calculator to determine the length of a circular arc. Simply input the radius of the circle and the central angle, specifying whether the angle is in degrees or radians.

units

Degrees Radians

function calculateArcLength() { var radiusInput = document.getElementById("radius").value; var centralAngleInput = document.getElementById("centralAngle").value; var angleUnit = document.getElementById("angleUnit").value; var resultDiv = document.getElementById("result"); // Input validation if (radiusInput === "" || centralAngleInput === "") { resultDiv.innerHTML = "Please enter values for both radius and central angle."; return; } var radius = parseFloat(radiusInput); var centralAngle = parseFloat(centralAngleInput); if (isNaN(radius) || isNaN(centralAngle)) { resultDiv.innerHTML = "Please enter valid numbers for radius and central angle."; return; } if (radius < 0) { resultDiv.innerHTML = "Radius cannot be negative."; return; } if (centralAngle < 0) { resultDiv.innerHTML = "Central angle cannot be negative."; return; } var arcLength; var angleInRadians; if (angleUnit === "degrees") { // Convert degrees to radians: radians = degrees * (PI / 180) angleInRadians = centralAngle * Math.PI / 180; } else { // radians angleInRadians = centralAngle; } // Arc Length Formula: s = r * θ (where θ is in radians) arcLength = radius * angleInRadians; resultDiv.innerHTML = "The length of the arc is: " + arcLength.toFixed(4) + " units."; }

Understanding Arc Length

An arc is a continuous portion of the circumference of a circle. The arc length is the distance along this curved segment. Calculating arc length is a fundamental concept in geometry and has applications in various fields, from engineering to architecture.

The Arc Length Formula

The most common formula to calculate the length of an arc is:

s = r × θ

Where:

  • s represents the arc length.
  • r is the radius of the circle.
  • θ (theta) is the central angle subtended by the arc, measured in radians.

Why Radians?

It's crucial that the central angle θ is in radians for the formula s = r × θ to work directly. A radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius. This definition makes the relationship between arc length, radius, and angle elegantly simple when using radians.

If your angle is given in degrees, you must first convert it to radians using the conversion factor:

Radians = Degrees × (π / 180)

Examples of Arc Length Calculation

Let's look at a couple of examples to illustrate the calculation:

Example 1: Angle in Degrees

Suppose you have a circle with a radius of 10 units and a central angle of 90 degrees.

  1. Convert degrees to radians:
    θ = 90 × (π / 180) = π / 2 radians
  2. Apply the arc length formula:
    s = r × θ = 10 × (π / 2) = 5π
  3. Approximate value:
    s &approx; 5 × 3.14159 &approx; 15.708 units

Example 2: Angle in Radians

Consider a circle with a radius of 5 units and a central angle of π/3 radians.

  1. The angle is already in radians:
    θ = π / 3 radians
  2. Apply the arc length formula:
    s = r × θ = 5 × (π / 3) = 5π / 3
  3. Approximate value:
    s &approx; (5 × 3.14159) / 3 &approx; 5.236 units

How to Use the Calculator

Our Arc Length Calculator simplifies this process for you:

  1. Enter the Radius: Input the numerical value of the circle's radius into the "Radius" field.
  2. Enter the Central Angle: Input the numerical value of the central angle into the "Central Angle" field.
  3. Select Angle Unit: Choose whether your central angle is in "Degrees" or "Radians" from the dropdown menu.
  4. Calculate: Click the "Calculate Arc Length" button.

The calculator will instantly display the arc length, making it easy to find the distance along any circular arc.

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