How to Calculate the Length of an Arc

Arc Length Calculator

Degrees (°) Radians (rad)

Results:

Arc Length: 0

function calculateArcLength() { var radius = parseFloat(document.getElementById('radius').value); var angle = parseFloat(document.getElementById('angle').value); var unit = document.getElementById('unit').value; var resultContainer = document.getElementById('result-container'); var resultSpan = document.getElementById('arcLengthValue'); var formulaSpan = document.getElementById('formulaUsed'); if (isNaN(radius) || isNaN(angle) || radius <= 0 || angle < 0) { alert("Please enter valid positive numbers for radius and angle."); return; } var arcLength; if (unit === 'degrees') { // Formula: L = (θ/360) * 2 * π * r arcLength = (angle / 360) * 2 * Math.PI * radius; formulaSpan.innerText = "Calculated using: (Angle / 360) × 2 × π × Radius"; } else { // Formula: L = r * θ arcLength = radius * angle; formulaSpan.innerText = "Calculated using: Radius × Angle (in Radians)"; } resultSpan.innerText = arcLength.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); resultContainer.style.display = 'block'; }

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How to Calculate the Length of an Arc

In geometry, an arc is a portion of the circumference of a circle. Calculating its length depends on two primary factors: the size of the circle (the radius) and how much of the circle the arc covers (the central angle).

The Arc Length Formulas

The method you use to find arc length depends on whether your central angle is measured in degrees or radians.

1. Using Degrees:
s = (θ / 360) × 2πr
Where θ is the angle in degrees, r is the radius, and π is approximately 3.14159.

2. Using Radians:
s = r × θ
This formula is much simpler, but the angle θ must be in radians.

Step-by-Step Calculation Example

Let's say you have a circle with a radius of 10 cm and a central angle of 60 degrees. Here is how you calculate the arc length:

1. Identify the variables: r = 10, θ = 60°.
2. Apply the degree formula: Arc Length = (60 / 360) × 2 × 3.14159 × 10.
3. Simplify the fraction: 60/360 is 1/6.
4. Multiply the terms: (1/6) × 62.8318.
5. Final Result: The arc length is approximately 10.47 cm.

Major vs. Minor Arcs

When two points are placed on a circle, they create two arcs. The smaller one is called the minor arc (less than 180°), and the larger one is called the major arc (greater than 180°). To find the length of a major arc if you only know the minor angle, simply subtract the minor angle from 360° before using the formula.

Common Applications

• Engineering: Designing curved roads, bridges, and mechanical gears.
• Astronomy: Calculating the distance between celestial bodies based on angular separation.
• Navigation: Determining distances along the Earth's curved surface (great-circle distances).