An arc is a portion of the circumference of a circle. The arc length is the distance along the curved line that makes up the arc. It's a fundamental concept in geometry and trigonometry, with applications in fields ranging from engineering and physics to computer graphics and navigation.
To calculate the arc length, we need two key pieces of information about the circle and the specific arc:
Radius (r): The distance from the center of the circle to any point on its circumference.
Central Angle (θ): The angle formed at the center of the circle by the two radii that connect the center to the endpoints of the arc. This angle can be measured in either radians or degrees.
The Formula
The formula for the arc length (s) depends on the unit of measurement for the central angle:
If the angle is in radians:
The formula is straightforward:
s = r * θ
where 's' is the arc length, 'r' is the radius, and 'θ' is the central angle in radians. This formula is derived from the definition of a radian, where one radian subtends an arc equal in length to the radius.
If the angle is in degrees:
First, we need to convert the angle from degrees to radians, or use a modified formula that accounts for the degree measure. To convert degrees to radians, we use the relationship π radians = 180 degrees. So, θ (radians) = θ (degrees) * (π / 180).
Substituting this into the radian formula gives:
s = r * (θ_degrees * π / 180)
This can also be expressed as:
s = (θ / 360) * 2πr
This form represents the arc length as a fraction of the circle's total circumference (2πr), where the fraction is determined by the ratio of the central angle to a full circle (360 degrees).
How to Use This Calculator
To find the arc length using this calculator:
Enter the Radius (r) of the circle.
Enter the measure of the Central Angle (θ).
Select the correct unit for your angle measurement (Radians or Degrees).
Click the "Calculate Arc Length" button.
The calculator will then display the calculated arc length, using the appropriate formula based on your unit selection.
Use Cases
Engineering: Calculating lengths for curved structures, road designs, or machine parts.
Physics: Analyzing circular motion, projectile paths, or the length of a pendulum's swing.
Geometry: Solving problems involving sectors, segments, and other parts of circles.
Computer Graphics: Rendering curved shapes and paths accurately on screen.
Architecture: Designing curved elements in buildings and landscapes.
function calculateArcLength() {
var radiusInput = document.getElementById("radius");
var angleInput = document.getElementById("angle");
var angleUnits = document.getElementById("angleUnits").value;
var resultDiv = document.getElementById("result");
var radius = parseFloat(radiusInput.value);
var angle = parseFloat(angleInput.value);
// Clear previous result and error messages
resultDiv.innerHTML = "";
// Input validation
if (isNaN(radius) || radius <= 0) {
resultDiv.innerHTML = "Please enter a valid positive number for the radius.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
if (isNaN(angle) || angle < 0) {
resultDiv.innerHTML = "Please enter a valid non-negative number for the angle.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
var arcLength;
var PI = Math.PI;
if (angleUnits === "radians") {
// Formula: s = r * θ (where θ is in radians)
arcLength = radius * angle;
} else if (angleUnits === "degrees") {
// Formula: s = (θ / 360) * 2πr OR s = r * (θ * π / 180)
// Convert degrees to radians first for consistency and clarity in calculation
var angleInRadians = angle * (PI / 180);
arcLength = radius * angleInRadians;
} else {
resultDiv.innerHTML = "Invalid angle unit selected.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
// Display result
if (!isNaN(arcLength)) {
resultDiv.innerHTML = "Calculated Arc Length: " + arcLength.toFixed(4) + "";
resultDiv.style.backgroundColor = getComputedStyle(document.documentElement).getPropertyValue('–success-green'); // Reset to green
} else {
resultDiv.innerHTML = "An error occurred during calculation.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
}
}