Ellipse Perimeter Calculator (Ramanujan)

Unlike a circle, an ellipse has no simple closed-form perimeter formula. The exact answer involves an “elliptic integral” — a numerical computation. For practical purposes, Ramanujan’s 1914 approximation is accurate to within 0.04% for any reasonable ellipse:

P ≈ π × [3(a + b) − √((3a + b)(a + 3b))]

Where a is the semi-major axis (half the longer diameter) and b is the semi-minor axis (half the shorter diameter).

For a circle (a = b = r), this reduces to 2πr, the familiar circumference formula. The further a and b are apart, the more elongated the ellipse and the harder it becomes to compute the exact perimeter — but Ramanujan’s formula stays accurate even for very elongated shapes.

Where ellipse perimeters matter in real life:

• Race tracks. A standard 400 m running track is two semicircles connected by two straights — not a true ellipse, but the inner lane geometry approximates one. Sport-track engineers use ellipse formulas to design banking curves.
• Whispering galleries (St Paul’s Cathedral in London, Mormon Tabernacle in Salt Lake City). The acoustic focus property comes from ellipse geometry — sound from one focus converges at the other.
• Oval gardens and ponds. A 12 ft × 8 ft oval pond perimeter ≈ π × [30 − √(22 × 18)] ≈ 31.7 ft of edging stones needed.
• Decorative oval mirrors. A 24 × 18 in oval mirror needs 67.4 in of frame molding.
• Astronomy. Planetary orbital paths are ellipses with the sun at one focus. Earth’s orbital perimeter is approximately 940 million km.
• Stadium and amphitheater designs for sightline calculations.

Worked example — oval pond:

A backyard pond, ellipse 16 ft × 10 ft. Semi-axes: a = 8 ft, b = 5 ft. P ≈ π × [3(13) − √(29 × 23)] ≈ π × [39 − √667] ≈ π × [39 − 25.83] ≈ π × 13.17 ≈ 41.4 ft.

For edging stones: order 45 linear ft to allow for fitting around the curve.

Worked example — oval rug binding:

Oval bedside rug, 5 × 3 ft. Semi-axes: a = 2.5 ft, b = 1.5 ft. P ≈ π × [3(4) − √(9 × 7)] ≈ π × [12 − √63] ≈ π × [12 − 7.94] ≈ π × 4.06 ≈ 12.75 ft of edge binding tape.

Eccentricity (how oval, vs how circular):

e = √(1 − b²/a²)

• e = 0: a perfect circle
• e = 0.5: moderate ellipse (like a typical decorative oval)
• e = 0.9+: highly elongated (cigar-shaped)
• e = 1: parabola (the limiting case, not actually a closed ellipse)

Earth’s orbital eccentricity is 0.0167 — barely measurable from a casual drawing. The orbit is nearly circular.

Why Ramanujan’s formula is so loved: it’s accurate, fast to compute, and uses only basic arithmetic (no integrals or transcendental functions beyond π and √). Ramanujan published several increasingly accurate approximations; this one is widely used because it hits the sweet spot of simplicity and accuracy.