Ellipse Area and Perimeter: Ovals for Gardens, Pools, and Tables
October 29, 2026
An ellipse is an oval shape defined by its semi-major axis a (half the longest width) and semi-minor axis b (half the shortest width). Its area is simple: A = π × a × b. For an oval garden bed 10 feet long and 6 feet wide, a = 5 and b = 3, so the area is π × 5 × 3 = 47.12 square feet. If a = b, it is a circle and the formula becomes πr².
The perimeter is another story. Unlike the circle, the ellipse has no simple closed-form formula; it requires an elliptic integral. For practical work, approximations are used.
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A rough approximation is P ≈ 2π√((a² + b²) ÷ 2). For a = 5 and b = 3, it gives 2π × √17 = 25.91. A more accurate one is Ramanujan's: with h = ((a − b) ÷ (a + b))², P ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))). For a = 5 and b = 3: h = (2 ÷ 8)² = 0.0625, and P ≈ π × 8 × (1 + 0.1875 ÷ (10 + 1.9764)) = 25.13 × 1.01566 = 25.53. This approximation is accurate to within a tiny fraction of a percent for most shapes. The simple approximation is 1.5% too high in this case.
A second Ramanujan approximation, P ≈ π[3(a + b) − √((3a + b)(a + 3b))], gives the same 25.53 for this ellipse.
| Semi-axes a, b | Area | Rough perimeter | Ramanujan perimeter |
|---|---|---|---|
| 5, 5 (circle) | 78.54 | 31.42 | 31.42 |
| 5, 4 | 62.83 | 28.45 | 28.36 |
| 5, 3 | 47.12 | 25.91 | 25.53 |
| 5, 2 | 31.42 | 23.93 | 23.01 |
| 5, 1 | 15.71 | 22.65 | 21.01 |
Oval Pools, Tables, and Beds
An oval pool 24 feet long and 12 feet wide (a = 12, b = 6) has a surface area of π × 12 × 6 = 226.2 square feet, and a perimeter of about 58.1 feet using Ramanujan's formula, which matters for the length of coping stones or a safety cover edge. With an average depth of 4.5 feet, the volume is 226.2 × 4.5 = 1,018 cubic feet, or 7,614 gallons. An oval dining table 72 by 42 inches has a = 36, b = 21 and an area of 2,375 square inches (16.5 square feet), seating about 6 to 8. An oval rug 5 by 8 feet has an area of π × 2.5 × 4 = 31.4 square feet, compared with 40 for a rectangle, and costs less in some materials.
A quick check for area: an ellipse fills about 78.5% (π/4) of its bounding rectangle.
Drawing an Ellipse
The string method, the 'gardener's ellipse', draws a true ellipse. Drive two stakes (the foci) into the ground, tie a loop of string around them, and trace around with a pencil or marker held taut against the string. The distance between the foci is 2c where c = √(a² − b²). For a bed with a = 5 and b = 3: c = 4, so the foci are 8 feet apart and the string loop must have a length of 2a + 2c = 10 + 8 = 18 feet. This works for beds, patios, and layouts at any scale.
Related Facts
The orbits of planets are ellipses with the Sun at one focus. Whispering galleries use the reflective property of ellipses: sound from one focus reaches the other. Eccentricity e = c ÷ a measures how stretched an ellipse is: 0.8 for the example, compared to 0 for a circle.
Frequently Asked Questions
What is the area of an ellipse?
π × a × b, with a and b the semi-axes.
Is there an exact formula for the perimeter?
No simple one. It involves an elliptic integral, so approximations like Ramanujan's are used.
How do I draw an ellipse on the ground?
Use two stakes and a loop of string of length 2a + 2c, tracing with a marker.
How much of its bounding box does an ellipse fill?
About 78.5%, equal to π/4.
What are the foci of an ellipse?
Two points inside it, at distance c = √(a² − b²) from the center along the major axis.