Arc Length Calculator

Enter radius and central angle for circular arcs, or input point coordinates in 2D or 3D space to compute precise arc lengths, chord lengths, sector areas, and spatial distances.
Luis GonzalezCreated by Luis GonzalezLast updated:

How to Use This Calculator

  1. 1

    Select Calculation Mode

    Choose 'Circular Arc (radius + angle)', 'Distance Between Two Points (2D)', or 'Distance Between Two Points (3D)' from the dropdown menu.

  2. 2

    Enter Circular Arc Parameters (if applicable)

    For circular arcs, enter the circle's radius and central angle in degrees.

  3. 3

    Enter Coordinate Values (if applicable)

    For 2D or 3D line segments, enter the start coordinates (x₁, y₁, z₁) and end coordinates (x₂, y₂, z₂).

  4. 4

    Review Comprehensive Geometric Metrics

    View exact arc length, chord length, sector area, sagitta, slope, or spatial Euclidean distance alongside contextual geometric insights.

Example Calculation

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

Radius

5

Central Angle (°)

90 °

Calculation Mode

Circular Arc (radius + angle)

Results

Arc Length

7.853982

Chord Length

7.071068

Sector Area

19.634954

Tips

Angles in Radians vs. Degrees

Analytical formulas use radians where θ = angle(°) × (π / 180). For a 90° angle, this converts to π/2 ≈ 1.570796 radians.

Arc Length vs. Chord Length

Arc length measures distance along the curved perimeter, whereas chord length is the direct line connecting the arc's endpoints. Arc length is always strictly greater than chord length for non-zero angles.

3D Distance in Spatial Engineering

When calculating paths in 3D space, ignoring the z-coordinate component (Δz) underestimates true distance. Ensure all three spatial axes are included for accurate physical measurements.

Unraveling Curved Paths and Distances: The Arc Length Calculator

Whether you are an architect designing a curved archway, an engineer surveying a curved roadway, or a student solving trigonometry problems, accurately measuring arc lengths and spatial distances is fundamental.

This calculator provides precision tools to compute circular arc lengths, chord lengths, sector areas, sagitta heights, and Euclidean distances across 2D and 3D coordinate systems in 2026.

For a circular arc with a 5-unit radius and a 90-degree central angle, the arc length is exactly 7.853982 units, the straight-line chord length is 7.071068 units, and the enclosed sector area is 19.634954 square units.

Core Mathematical Formulas

The calculator applies different geometric principles depending on the active mode:

1. Circular Arc (Radius + Angle)

Angle in Radians (θ) = Angle (degrees) × (π / 180)
Arc Length (s)        = Radius (r) × θ
Chord Length (c)      = 2 × r × sin(θ / 2)
Sector Area (A)       = 0.5 × r² × θ
Sagitta (h)           = r × (1 - cos(θ / 2))
Full Circumference    = 2 × π × r

2. Straight-Line Distance (2D)

Distance (d) = √((x₂ - x₁)² + (y₂ - y₁)²)
Incline Angle = arctan(|y₂ - y₁| / |x₂ - x₁|) × (180 / π)
Midpoint      = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

3. Spatial Distance (3D)

Distance (d) = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
XY Projection = √((x₂ - x₁)² + (y₂ - y₁)²)
Elevation Angle = arctan(|z₂ - z₁| / XY Projection) × (180 / π)

Worked Example: Step-by-Step Arc Calculation

Consider calculating all metrics for a circular segment with radius r = 5 and central angle θ = 90°:

  1. Convert Angle to Radians: $$\theta = 90^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{2} \approx 1.570796327 \text{ rad}$$

  2. Calculate Arc Length: $$s = 5 \times 1.570796327 = 7.853982 \text{ units}$$

  3. Calculate Chord Length: $$c = 2 \times 5 \times \sin\left(\frac{1.570796327}{2}\right) = 10 \times \sin(0.785398163) = 10 \times 0.707106781 = 7.071068 \text{ units}$$

  4. Calculate Sector Area: $$A = 0.5 \times 5^2 \times 1.570796327 = 0.5 \times 25 \times 1.570796327 = 19.634954 \text{ sq units}$$

  5. Calculate Sagitta (Rise): $$h = 5 - 5 \times \cos(0.785398163) = 5 - 5 \times 0.707106781 = 1.464466 \text{ units}$$

  6. Calculate Total Circumference: $$C = 2 \times \pi \times 5 = 31.415927 \text{ units}$$

Real-World Applications in Engineering & Design

Arc length calculations are critical across multiple professional disciplines:

  • Civil Engineering: Surveying horizontal curve alignments for roadways, highway exit ramps, and railway curves.
  • Architecture: Dimensioning structural stone arches, curved glass facades, and dome structures.
  • Manufacturing & Fabrication: Determining sheet metal unroll lengths before bending or forming cylindrical parts.
  • Robotics & Navigation: Calculating curved wheel path trajectories and toolhead travel distances on 3D CNC machines.

Frequently Asked Questions

What is the difference between arc length and chord length?

Arc length is the distance along the curved perimeter of a circle segment (s = rθ). Chord length is the straight-line distance connecting the arc's two endpoints (c = 2r sin(θ/2)). For a radius of 5 and central angle of 90°, the arc length is 7.853982 units while the straight chord length is 7.071068 units.

How is sector area calculated?

Sector area is the total two-dimensional space enclosed by the arc and the two radii extending to its endpoints. It is calculated as A = 0.5 × r² × θ (with θ in radians). For a 90° angle (π/2 radians) and radius of 5, the sector area is 0.5 × 25 × 1.570796 = 19.634954 square units, representing exactly 25% of the total circle area.

What is the sagitta of an arc?

The sagitta (or height/rise) is the distance from the midpoint of the chord to the midpoint of the arc, given by h = r × (1 - cos(θ/2)). For a circle with radius 5 and central angle of 90°, the sagitta is 1.464466 units.

How does 2D Euclidean distance differ from 3D spatial distance?

2D Euclidean distance uses the Pythagorean theorem √((x₂ - x₁)² + (y₂ - y₁)²). 3D distance incorporates depth or height via √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²).