Ellipse Area Calculator

An ellipse is a stretched circle. Two semi-axes define it: a (longer, semi-major) and b (shorter, semi-minor).

A = π × a × b

If a = b, the ellipse is a circle and the formula reduces to π × r². The further apart a and b are, the more elongated the oval.

Perimeter is harder. Unlike area, there is no simple closed-form formula. Ramanujan’s well-known approximation gives 4-decimal accuracy for moderate eccentricities:

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

This is the formula this calculator uses for the perimeter readout.

Where ellipses appear in measurement:

• Running tracks. A standard 400-meter track has two 100 m straights and two 100 m semicircles, but lane 1 measured to the inside edge isn’t exactly semicircular at the corners — most modern tracks use a flattened-curve shape that approximates an ellipse to keep the geometry consistent across lanes.
• Whispering galleries (St Paul’s Cathedral, the US Capitol) are ellipsoidal. Sound waves from one focus converge at the other — a fact that’s a direct consequence of how the geometry is defined.
• Orbits. Planet orbits are ellipses with the sun at one focus. Earth’s orbital semi-major axis is 149.6 million km; eccentricity is only 0.0167, so it’s nearly circular.
• Mirrors and reflectors. Oval bathroom mirrors and decorative wall art are usually ellipses.
• Football and rugby ball cross-section. Roughly elliptical, allowing predictable spiral behavior.

Worked example — oval rug:

A 6 ft × 9 ft oval rug. Semi-axes are a = 4.5 ft, b = 3 ft. Area = π × 4.5 × 3 ≈ 42.4 sq ft. Perimeter (Ramanujan): π × [22.5 − √(16.5 × 13.5)] ≈ π × [22.5 − 14.92] ≈ 23.8 ft.

If you were edging the rug with a fabric band, you’d need about 24 ft of trim — slightly less than the 24 ft perimeter of the bounding 6 × 9 rectangle.

Worked example — pond design:

Backyard pond designed as an ellipse 12 ft × 8 ft. Semi-axes 6 ft and 4 ft. Area = π × 6 × 4 ≈ 75.4 sq ft. Perimeter ≈ π × [30 − √(22 × 18)] ≈ π × [30 − 19.90] ≈ 31.7 ft (useful for sizing edging stones).

Sanity check: the bounding rectangle 12 × 8 has area 96 sq ft. The ellipse uses π/4 ≈ 78.5% of that — that’s the constant ratio of an ellipse’s area to its bounding rectangle, just like a circle uses π/4 of its bounding square.

Eccentricity (how oval the ellipse is) is e = √(1 − b²/a²). Earth’s orbit: e = 0.0167 (very nearly circular). A 2:1 ratio ellipse (a = 2b): e = √(0.75) ≈ 0.866. The closer to 1, the more elongated.