How To Calculate An Arc Between Two Points

Calculate arc length, central angle, and chord length from coordinates, radius, or angle inputs.

How to Calculate an Arc Between Two Points: Complete Expert Guide

Calculating an arc between two points is a core geometry skill that appears in engineering, CAD modeling, surveying, transportation design, robotics, and geospatial analysis. At a basic level, an arc is a curved part of a circle connecting two endpoints. If you know enough geometry about the circle, especially the radius or central angle, you can compute the exact arc length quickly and reliably.

Many people confuse arc length with straight line distance. The straight line distance between the same two points on a circle is called the chord length, not the arc length. The arc is always at least as long as the chord, and for larger angles the difference can be substantial. Knowing when to use each measure can prevent major design and estimation errors.

Core Definitions You Must Know

• Chord (c): straight line distance between two points on a circle.
• Radius (r): distance from center of the circle to any point on the circumference.
• Central angle (θ): angle formed at the center by the two radii connecting to the points.
• Arc length (s): curved distance along the circle from point A to point B.
• Minor arc: the shorter arc between two points.
• Major arc: the longer arc between the same points.

The Most Important Formula

The standard arc length formula is:

s = r × θ (when θ is in radians)

If angle is in degrees, convert first:

θ_rad = θ_deg × π / 180

Then compute:

s = r × θ_rad

When You Only Know Two Points and Radius

In practical design, you often know the coordinates of two points and the circle radius. In that case:

1. Calculate chord length: c = √((x2 – x1)² + (y2 – y1)²)
2. Check feasibility: c ≤ 2r. If not, those points cannot lie on the same circle with radius r.
3. Compute minor central angle: θ_minor = 2 asin(c / (2r))
4. Minor arc: s_minor = r × θ_minor
5. Major arc: s_major = r × (2π – θ_minor)

Practical tip: if your workflow includes GIS, road curvature, or machine motion planning, always store both chord and arc values. Chord helps with direct distance checks, arc helps with travel path and material estimates.

Step by Step Numerical Example

Assume A(0,0), B(6,0), radius r = 5.

1. Chord: c = 6
2. Feasibility: 6 ≤ 10, valid
3. θ_minor = 2 asin(6 / 10) = 2 asin(0.6) ≈ 1.287 radians
4. s_minor = 5 × 1.287 = 6.435 units
5. s_major = 5 × (2π – 1.287) ≈ 24.981 units

This example highlights why specifying minor versus major arc matters. Same endpoints, very different path lengths.

Arc vs Chord Difference Statistics

The table below shows how arc length differs from chord length for the same radius at common central angles. These values come directly from exact geometry relations and illustrate why chord approximation becomes risky at wider angles.

Central Angle (deg)	Arc/Chord Ratio	Arc Longer Than Chord	Interpretation
10	1.00127	0.127%	Almost identical for very small arcs
30	1.0115	1.15%	Small but noticeable in precision work
60	1.0472	4.72%	Meaningful underestimation if chord is used
90	1.1107	11.07%	Large error for quarter circle designs
120	1.2092	20.92%	Chord is not acceptable proxy in most projects
180	1.5708	57.08%	Semicircle arc is much longer than diameter

Error Sensitivity and Unit Discipline

Arc length is linearly proportional to radius and angle. This gives a very clear error rule:

• If radius has 1% error and angle is exact, arc has 1% error.
• If angle has 1% error and radius is exact, arc has 1% error.
• If both have errors, total uncertainty can compound.

Unit consistency is also critical. If radius is entered in meters and you label output as feet, your result is numerically correct but physically wrong for your project context. Always standardize your unit system before calculating.

Input Issue	Typical Magnitude	Resulting Arc Impact	Best Practice
Degrees used as radians	Up to 57.3x scaling error	Catastrophic overestimate or underestimate	Convert deg to rad before applying s = rθ
Radius measurement uncertainty	0.5% to 2%	Same percent error in arc length	Use calibrated measurements
Rounding angle too early	0.1 deg to 1 deg	Visible error on large radii	Keep at least 4 to 6 decimal places in radians
Using chord as arc	0.1% to 57%+ depending on angle	Systematic underestimation	Use arc formula, reserve chord for straight distance only

Common Real World Use Cases

• Road and rail geometry: transition curves and circular segments require precise arc lengths for safety and comfort.
• Piping and fabrication: bend development often depends on arc lengths rather than straight dimensions.
• CNC and robotics: toolpaths frequently include circular interpolation where arc distance controls feed timing.
• GIS and geodesy: Earth distances can be modeled as spherical arcs for regional scale planning.
• Architecture: curved facades and domes need accurate material takeoff along arcs.

Advanced Note: Planar Arc vs Earth Surface Arc

This calculator assumes a flat 2D circle in Euclidean geometry. For Earth-scale distances, you may need spherical or ellipsoidal geodesic methods. A surface route between two latitude and longitude points is an arc on a sphere-like body, not a flat circle in a plane. The principle is similar, but models and formulas differ, especially over long distances.

If you work in mapping or surveying, review geodetic resources and standards from national agencies and university engineering programs before selecting a distance model for compliance-grade projects.

Authoritative Learning Resources

• MIT OpenCourseWare (Calculus and arc length fundamentals)
• NOAA National Geodetic Survey (geodesy and Earth measurement)
• NASA Planetary Fact Sheets (reference radii and planetary geometry data)

Final Checklist for Accurate Arc Calculations

1. Identify whether you need minor or major arc.
2. Use consistent units for every input.
3. Convert angles to radians before multiplying by radius.
4. If using two points plus radius, verify c ≤ 2r.
5. Report chord, central angle, and arc together for transparency.
6. For large geographic distances, switch to geodesic models.

With these methods, you can compute arcs between two points with confidence, avoid common geometry mistakes, and produce results suitable for technical documentation and professional engineering workflows.