Ellipse Arc Length Calculator: Formula & Steps

This calculator finds the arc length of an ellipse when you enter the semi-major axis (a), the semi-minor axis (b), and the angle span (from 0 to a chosen upper angle, in degrees). It returns the arc length in the same units you used for the axes, using the standard complete elliptic integral of the second kind for a full perimeter or a numerical approximation for partial arcs.

Formula

For a full ellipse (angle span = 360°), the perimeter P is given by the complete elliptic integral of the second kind:

P = 4a × E(e)

where e is the eccentricity, e = √(1 − (b/a)²), and E(e) is the complete elliptic integral. Because E(e) has no elementary closed form, we use a widely accepted approximation (Ramanujan’s or a series expansion). The calculator uses a series approximation:

E(e) ≈ 1 − (1/4)e² − (3/64)e� − (5/256)e� − …

For a partial arc from 0 to angle θ (in degrees), the arc length is proportionally scaled: L = P × (θ / 360). This works exactly only for circular arcs; for ellipses it is an approximation that is quite good for small eccentricities and moderate angles, but not exact for large arcs.

Variables:

• a = semi-major axis (same unit as output, e.g., meters, feet)
• b = semi-minor axis (same unit)
• θ = angle span in degrees (0 to 360)
• L = arc length (same unit as a and b)

Worked Example

Using the default values: a = 5 m, b = 3 m, angle = 360°.

1. Compute eccentricity squared: e² = 1 − (b/a)² = 1 − (3/5)² = 1 − 0.36 = 0.64.
2. Compute the series: 1 − 0.64*(1/4) − 0.64²*(3/64) − 0.64³*(5/256) = 1 − 0.16 − 0.0192 − 0.005 = 0.8158.
3. Perimeter = 4 × 5 × 0.8158 = 16.316 m.
4. Angle fraction = 360/360 = 1, so arc length = 16.316 m.

The calculator returns 16.316 m. For a partial arc, say 90°, you would get 16.316 × 0.25 = 4.079 m.

Why Ellipse Arc Length Is Hard

Unlike a circle, where arc length is a simple fraction of the circumference, an ellipse’s perimeter cannot be expressed with basic functions. The integral that defines it—the elliptic integral of the second kind—has no closed-form antiderivative. This is why you see approximations or numerical methods in calculators. The series used here converges quickly for moderate eccentricities (e < 0.9), which covers most real-world ellipses like orbits, arches, and ellipsoidal tanks.

How to Use the Calculator Correctly

Enter the semi-major axis (the longer radius) and semi-minor axis (the shorter radius). Ensure both are in the same unit—if you mix meters and feet, the result will be meaningless. The angle span is measured from the positive x-axis (the direction of the semi-major axis) counterclockwise. For a full ellipse, use 360°. For a partial arc, enter the desired angle in degrees (0–360). The result is the length along the curve, not the straight-line chord distance.

Common use cases:

• Architecture: length of an elliptical arch or dome rib.
• Engineering: length of an elliptical pipe or duct cross-section.
• Astronomy: approximate distance along an elliptical orbit (but note that orbital speed varies, so time is not proportional to angle).

Common Mistakes and Limitations

Mistakes often come from units and angle interpretation:

• Mixing units (e.g., a in meters, b in feet) gives a wrong result.
• Using the diameter instead of the semi-axis (halve the diameter first).
• Assuming the arc length scales linearly with angle for large eccentricities. The proportional scaling is exact only for circles; for ellipses it overestimates or underestimates depending on the starting angle. For a full ellipse it is exact, but for partial arcs it is an approximation.

Limitations: The series approximation loses accuracy for very high eccentricities (e > 0.95). For such cases, use numerical integration. Also, the proportional scaling for partial arcs is not accurate for arcs that span more than about 90° on a highly eccentric ellipse. For precise work, use a numerical integrator.

Alternatives and More Accurate Methods

If you need exact arc length for any partial ellipse, use numerical integration of the arc length formula:

L = ∫ √(a² sin² t + b² cos² t) dt from t1 to t2

where t is the parametric angle (not the polar angle). Many graphing calculators and software like Python (scipy) or MATLAB can compute this. For full perimeter, Ramanujan’s approximation is remarkably accurate:

P ≈ π [3(a+b) − √((3a+b)(a+3b))]

That formula is accurate to within about 0.04% for most ellipses, but the series used here is fine for typical cases.

Frequently Asked Questions

Q: Is the arc length of an ellipse proportional to the angle?
No, only for a circle. For an ellipse, the arc length per degree varies with position. The proportional method used here is a reasonable approximation for small arcs or low eccentricity, but for exact values you need numerical integration.

Q: What is the difference between the parametric angle and the polar angle?
The parametric angle (t) is used in the ellipse equations x = a cos t, y = b sin t. The polar angle (θ) is measured from the center in the standard x-y plane. They differ unless the ellipse is a circle. Arc length formulas typically use the parametric angle.

Q: Can I use this calculator for an elliptical orbit segment?
Yes, but remember that the arc length is the distance along the path, not the time. Orbital speed varies, so you cannot derive time from angle alone. Also, the semi-major axis and semi-minor axis must be in the same units (e.g., kilometers).

Q: Why does my manual calculation differ from the calculator?
Check if you used degrees vs. radians, or if you mixed up semi-axes. Also, the calculator uses a series approximation; for high eccentricity, manual numerical integration may give a slightly different value. For most practical purposes, the difference is negligible.

Q: What is the eccentricity of an ellipse?
Eccentricity e = √(1 − (b/a)²). It ranges from 0 (circle) to just below 1 (very elongated). The higher the eccentricity, the more the arc length deviates from the simple proportional rule.

Related

• Area Calculator in Sq Ft: How to Use It
• Area Calculator in mm: Square Millimeters Explained
• Area Calculator in Sqm: How to Convert Any Shape

in a | Semi-major axis (a) | m | 5 | 0
in b | Semi-minor axis (b) | m | 3 | 0
in ang | Angle span (degrees) | deg | 360 | 0
out L | Arc length | m | a * (ang/360) * 4 * (1 - (1 - (b/a)^2) * (1/4 + (3/64)*(1 - (b/a)^2) + (5/256)*(1 - (b/a)^2)^2)) | 4