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, which look like ellipses and are not. A 400 m track is a stadium shape, straights joined by semicircles, and it is not made of 100 m pieces either. Lane 1 measured 30 cm off the kerb is two straights of 84.39 m and two bends of 115.61 m, which is what makes the lap come to exactly 400. Work one out on the stadium shape page, not this one.
• Whispering galleries. Sound leaving one focus of an ellipse converges on the other, which is the effect in Statuary Hall at the US Capitol. St Paul’s in London gets cited for this constantly and should not be: its gallery is circular, and the whisper travels around the wall rather than across between two foci.
• 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: an 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 would need about 24 ft of trim. The 6 × 9 rectangle it fits inside has a perimeter of 2(6 + 9) = 30 ft, so rounding the corners off saves you a fifth of the binding as well as a fifth of the fabric.

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, and the ratio of any ellipse’s area to its bounding rectangle is that same constant, just as 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.