Arc Length Calculator

Enter a radius and central angle for circular arcs, or input two points in 2D or 3D space to compute arc length and related geometric metrics.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Select Calculation Mode
Choose between 'Circular Arc (radius + angle)', 'Distance Between Two Points (2D)', or 'Distance Between Two Points (3D)'.
2
Enter Circular Arc Inputs (if applicable)
For circular arcs, input the radius of the circle and the central angle in degrees.
3
Enter 2D Point Coordinates (if applicable)
For 2D distance, input the x and y coordinates for both the start and end points.
4
Enter 3D Point Coordinates (if applicable)
For 3D distance, input the x, y, and z coordinates for both the start and end points.
5
Review your results
The calculator will display the arc length, chord length, sector area, or Euclidean distance based on your chosen mode.

Example Calculation

A mathematician needs to find the arc length, chord length, and sector area of a circular arc with a radius of 5 units and a central angle of 90 degrees.

Radius

Central Angle (°)

90 °

Calculation Mode

Circular Arc (radius + angle)

Results

7.853982

Tips

Convert Degrees to Radians for Formulas

Many mathematical formulas for arc length and sector area require the central angle to be in radians. Remember that 180 degrees equals π radians. For a 90-degree angle, this means 90 * (π/180) = π/2 radians.

Distinguish Arc Length from Chord Length

Arc length measures the distance along the curve, while chord length measures the straight-line distance between the two endpoints of the arc. The arc length will always be greater than or equal to the chord length (equal only for a 0° or 360° arc).

Use 3D Distance for Spatial Problems

When dealing with real-world objects or paths that move through three-dimensional space, always use the 3D distance calculation. Neglecting the z-coordinate in such cases will lead to an underestimation of the true path length or separation.

Unraveling Curved Paths and Distances: The Arc Length Calculator

Whether you're an engineer designing a curved road, an astronomer mapping celestial orbits, or a student solving geometry problems, precisely measuring arc lengths and distances is fundamental. This calculator offers versatile tools to compute circular arc lengths, chord lengths, and Euclidean distances in both two and three dimensions. For a circular arc with a 5-unit radius and a 90-degree central angle, the arc length is exactly 7.853982 units, illustrating the precise nature of these geometric calculations.

The Mathematical Principles of Arc Length and Distance

The Arc Length Calculator operates on distinct mathematical principles depending on the selected mode. For circular arcs, it leverages the relationship between a circle's radius and its central angle. For distances between points, it applies the Euclidean distance formula, an extension of the Pythagorean theorem.

For Circular Arc (Radius + Angle):

angle in radians = central angle (degrees) × (π / 180)
arc length = radius × angle in radians
chord length = 2 × radius × SIN(angle in radians / 2)
sector area = 0.5 × radius² × angle in radians

For Distance Between Two Points (2D):

distance = √((end X - start X)² + (end Y - start Y)²)

For Distance Between Two Points (3D):

distance = √((end X - start X)² + (end Y - start Y)² + (end Z - start Z)²)

These formulas allow for accurate measurement of both curved and straight paths in various spatial contexts.

💡 Understanding geometric relationships is essential in many fields. For optimizing space, our Container Utilization Percentage Calculator uses similar spatial reasoning.

Worked Example: Calculating a Circular Arc's Properties

Let's illustrate with a classic geometry problem: finding the properties of a circular arc.

Calculation Mode: Circular Arc (radius + angle)
Radius: 5 units
Central Angle: 90 degrees

First, convert the central angle to radians: angle in radians = 90° × (π / 180°) = π / 2 radians ≈ 1.570796 radians

Next, calculate the arc length: arc length = 5 × 1.570796 = 7.85398 units

Then, calculate the chord length: chord length = 2 × 5 × SIN(1.570796 / 2) = 10 × SIN(0.785398) = 10 × 0.707107 ≈ 7.07107 units

Finally, calculate the sector area: sector area = 0.5 × 5² × 1.570796 = 0.5 × 25 × 1.570796 = 19.63495 units²

The arc length is approximately 7.85398 units.

💡 For other mathematical concepts involving ratios and proportions, our Conversion Rate Percentage Calculator can help you analyze changes between values.

Arc Length in Geometry and Engineering Applications

Arc length is a fundamental concept with widespread applications across geometry, engineering, and various scientific disciplines. In civil engineering, it's crucial for designing and surveying curved sections of roads, railways, and pipelines, ensuring smooth transitions and accurate material estimates. Architects use arc length to define the curvature of arches, domes, and other architectural elements, impacting both structural integrity and aesthetic appeal. In physics, understanding arc length is essential for calculating the path of objects in circular or elliptical motion, such as planetary orbits or projectile trajectories. For instance, determining the length of a segment of a bridge arch with a 100-foot radius and a 60-degree central angle involves a precise arc length calculation (approximately 104.7 feet), illustrating its practical importance in real-world construction and design. Radians are the preferred unit for angular measurement in these advanced applications due to their direct relationship with arc length.

Standard Units and Conventions for Arc Measurement

The measurement of arcs and angles adheres to specific standard units and conventions to ensure clarity and consistency across scientific, engineering, and mathematical fields. The two primary units for angles are degrees and radians. Degrees, with a full circle being 360°, are intuitive for everyday use and navigation. Radians, however, are the standard in advanced mathematics and physics because they establish a direct, unitless relationship between arc length and radius (arc length = radius × angle in radians). This means that for a given radius, an angle of 1 radian subtends an arc exactly equal in length to the radius. International standards organizations, such as the International Organization for Standardization (ISO), promote the use of SI units, which include radians for plane angles, to facilitate global scientific communication and technical interoperability. This adherence to consistent units prevents errors and ensures that calculations are universally understood and reproducible.

Frequently Asked Questions

What is arc length in geometry?

Arc length is the distance along the curved line segment of a circle's circumference. It represents a portion of the total circumference, determined by the radius of the circle and the central angle that subtends (or 'cuts off') that arc. The arc length is measured in units of length, such as inches, centimeters, or meters, and is distinct from the straight-line distance between the arc's endpoints, known as the chord length.

How is arc length calculated for a circular arc?

For a circular arc, the arc length (s) is calculated using the formula s = rθ, where 'r' is the radius of the circle and 'θ' (theta) is the central angle subtended by the arc, measured in radians. If the angle is given in degrees, it must first be converted to radians using the conversion factor: radians = degrees × (π/180). This formula directly relates the size of the circle and the angle of the arc to its length.

What is the difference between Euclidean distance in 2D and 3D?

Euclidean distance measures the shortest straight-line path between two points. In 2D, it calculates the distance between two points (x₁, y₁) and (x₂, y₂) using the Pythagorean theorem: √((x₂-x₁)² + (y₂-y₁)²). In 3D, it extends this concept to include a z-coordinate, calculating the distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) as √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²), accounting for depth or height.