About the Ellipse Calculator
This ellipse calculator takes the semi-major axis (half the longest width) and the semi-minor axis (half the shortest width) and returns the area, the perimeter, the eccentricity, the distance from the centre to each focus and the semi-latus rectum. You can enter the two axes in either order — the larger one is always treated as the semi-major axis.
It is useful for students checking conic-section homework, and for practical jobs such as sizing an oval table top, rug, mirror, garden bed or racetrack, where you need the surface area for material or the perimeter for edging and trim.
The area formula πab is exact. The perimeter of an ellipse has no simple closed form, so the calculator uses Ramanujan’s second approximation, which is exact for a circle, accurate to better than one part in ten million for a 3 : 1 oval and still within about 0.001% for an ellipse ten times longer than it is wide.
With the default inputs, the area is 47.1239. Change any value above to recalculate instantly.
How to use the ellipse calculator
- 1Measure the full length and full width of the ellipse through its centre.
- 2Halve each to get the semi-major axis a and semi-minor axis b.
- 3Enter a and b in the same unit (either order works).
- 4Read the area in square units and the perimeter in linear units.
- 5Use eccentricity and focal distance for conic-section or orbit problems.
Formula and method
The area of an ellipse is exactly π times the product of the two semi-axes — the circle formula πr² with r replaced by a and b. The perimeter has no elementary closed form (it is an elliptic integral), so we use Ramanujan’s second approximation, whose error is negligible for practical shapes and zero for a circle.
Eccentricity measures how stretched the ellipse is: 0 is a circle and values near 1 are long and thin. The foci lie on the major axis at distance c = √(a² − b²) from the centre; the semi-latus rectum b²/a is the half-width of the ellipse measured through a focus.
- a
- Semi-major axis (the larger half-width)
- b
- Semi-minor axis (the smaller half-width)
- h
- Shape parameter used in Ramanujan’s perimeter formula
- e
- Eccentricity
- c
- Distance from the centre to each focus
Worked examples
Ellipse with a = 5 and b = 3
Area = π × 5 × 3 ≈ 47.12 square units. With h = (2/8)² = 0.0625, Ramanujan’s formula gives a perimeter of about 25.53. The foci are √(25 − 9) = 4 units from the centre, so the eccentricity is 4/5 = 0.8.
Equal axes (a circle of radius 10)
When a = b the ellipse is a circle, so the area is π × 10² ≈ 314.16 and the perimeter is 2π × 10 ≈ 62.83. Both foci collapse onto the centre and the eccentricity is 0.
Oval table top 24 × 8 (a = 12, b = 4)
A long oval 24 wide and 8 deep covers π × 12 × 4 ≈ 150.8 square units and needs about 53.46 units of edge banding. Its eccentricity of about 0.943 shows it is quite elongated.
Frequently asked questions
What is the formula for the area of an ellipse?+
Area = π × a × b, where a and b are the semi-major and semi-minor axes (half the length and half the width). For a 10 × 6 oval, the area is π × 5 × 3 ≈ 47.12 square units.
Why is there no exact formula for the perimeter of an ellipse?+
The perimeter is an elliptic integral that cannot be written with elementary functions. Approximations such as Ramanujan’s are used instead; his second formula is within about 0.000003% for a 3 : 1 ellipse and about 0.001% even for a 10 : 1 ellipse, far more precise than any tape measure.
What does eccentricity mean for an ellipse?+
Eccentricity e = c/a describes how stretched an ellipse is. A circle has e = 0; the closer e gets to 1, the flatter the ellipse. Earth’s orbit has e ≈ 0.017, which is almost circular.
Where are the foci of an ellipse?+
Both foci lie on the major axis, a distance c = √(a² − b²) from the centre. For any point on the ellipse, the sum of its distances to the two foci is always 2a.
Is an oval the same as an ellipse?+
An ellipse is a precise mathematical oval. Many everyday “ovals” (eggs, racetracks, stadium shapes) are not true ellipses, so this calculator is exact only for elliptical shapes; a stadium shape is a rectangle plus two semicircles.