Enter the semi-major axis a and semi-minor axis b (or the full width and height) to get the area, perimeter, eccentricity, focal distance, foci and the standard equation, with every step worked out.
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If b is larger than a they are swapped automatically.
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How to use the Ellipse Calculator
Choose semi-axes or full axes. Full axes are the overall length and width of the oval.
Enter a and b and pick a unit.
Optionally set the orientation and center to get coordinates of the foci and the equation.
Read the area and perimeter, then the eccentricity, foci and other properties.
Check the steps and the perimeter formula comparison, or copy the results.
Ellipse formulas
Property
Formula
Example (a = 5, b = 3)
Area
A = π·a·b
47.1239
Perimeter (Ramanujan II)
π(a + b)(1 + 3h / (10 + √(4 − 3h))), h = (a − b)²/(a + b)²
25.5270
Perimeter (Ramanujan I)
π[3(a + b) − √((3a + b)(a + 3b))]
25.5270
Focal distance
c = √(a² − b²)
4
Eccentricity
e = c / a = √(1 − b²/a²)
0.8
Latus rectum
2b² / a
3.6
Flattening
f = (a − b) / a
0.4
Directrix
x = ± a / e
± 6.25
The defining property of an ellipse is that the sum of the distances from any point on the curve to the two foci is constant and equals the major axis 2a. That is why you can draw one with a loop of string around two pins.
Why the perimeter needs an approximation
Unlike a circle, an ellipse has no elementary formula for its circumference. The exact value is the elliptic integral P = 4a ∫0π/2 √(1 − e² sin²θ) dθ, which the calculator evaluates numerically. Srinivasa Ramanujan's 1914 approximations are so close that for everyday shapes the difference is invisible:
Axis ratio b/a
Eccentricity
Ramanujan II relative error
2π√((a²+b²)/2) relative error
0.9
0.436
< 0.0000001%
+0.07%
0.6
0.8
< 0.0000001%
+1.49%
0.3
0.954
about −0.000007%
+5.76%
0.1
0.995
about −0.001%
+9.87%
Frequently asked questions
How do you find the area of an ellipse?
Multiply π by the two semi-axes: A = π × a × b. For a = 5 and b = 3 the area is π × 15 ≈ 47.12 square units. If you know the full length and width instead, halve them first.
Is there an exact formula for the perimeter of an ellipse?
No simple one. The exact perimeter is 4a·E(e), where E is the complete elliptic integral of the second kind. Ramanujan's approximation P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))), with h = (a − b)² / (a + b)², is accurate to better than a few parts per million for most ellipses.
What is the eccentricity of an ellipse?
Eccentricity e = c / a = √(1 − b²/a²) measures how stretched the ellipse is. It is 0 for a circle and approaches 1 as the ellipse becomes very long and thin. Earth's orbit has e ≈ 0.0167, so it is almost circular.
How do I find the foci of an ellipse?
Compute c = √(a² − b²). The foci lie on the major axis at a distance c from the center, so for a horizontal ellipse centered at (h, k) they are at (h ± c, k). For a = 5 and b = 3, c = 4 and the foci are at (±4, 0).
What is the difference between the semi-major and semi-minor axis?
The semi-major axis a is half of the longest diameter, and the semi-minor axis b is half of the shortest diameter. They meet at right angles at the center. When a = b the ellipse is a circle with radius a.
Is an oval the same as an ellipse?
Not exactly. An ellipse is a precise curve defined by its two foci, while oval is a looser everyday word for any egg-like rounded shape. Many oval tables, rugs and mirrors are true ellipses, so this calculator works for them if you measure the overall length and width.
What is the standard equation of an ellipse?
For a horizontal ellipse centered at (h, k): (x − h)²/a² + (y − k)²/b² = 1. For a vertical one, a² goes under the y term: (x − h)²/b² + (y − k)²/a² = 1.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.