Ellipse Area Calculator
Type the two semi-axes of your ellipse and the calculator returns the area, an accurate perimeter approximation, the eccentricity and the focal distance.
Area (A = π·a·b)
47.12389 square units
- Semi-major axis (a)
- 5
- Semi-minor axis (b)
- 3
- Perimeter (Ramanujan)
- 25.53
- Eccentricity (e)
- 0.8
- Focal distance (c)
- 4
The area of an ellipse is π·a·b, where a and b are the semi-major and semi-minor axes. It is the circle area π·r² generalised: a circle is the special case a = b = r. The perimeter has no exact closed form, so the calculator uses Ramanujan's second approximation, accurate to better than one part in a million for any eccentricity. Eccentricity e = √(1 − b²/a²) measures how stretched the ellipse is (0 = circle, →1 = very elongated). Focal distance c = √(a² − b²) is the distance from the centre to each focus.
How to use this calculator
Enter the semi-major axis (a) and the semi-minor axis (b) — these are the two "half-widths" of the ellipse, from the centre to the edge along the longest and shortest directions. The unit you use does not matter as long as both values are in the same unit: centimetres, metres, inches or feet all work. The area comes back in the squared unit; perimeter and focal distance come back in the same linear unit. The calculator automatically sorts which axis is longer, so it does not matter whether you label them in the "right" order.
How the calculation works
The area of an ellipse is exactly A = π·a·b. This generalises the circle formula A = π·r²: a circle is just an ellipse with a = b = r. Unlike the area, the perimeter has no closed-form elementary expression — it is an elliptic integral. The calculator uses Ramanujan's second approximation P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))), where h = ((a − b)/(a + b))². This is accurate to roughly one part in 10^7 even for very elongated ellipses, which is more than enough for any drawing, machining, or design purpose. Eccentricity is e = √(1 − b²/a²); it is zero for a circle and approaches one as the ellipse stretches. The two foci sit on the major axis at distance c = √(a² − b²) from the centre — that is the famous "string and two pins" property used to draw ellipses by hand.
Worked example
Take an ellipse with a = 5 and b = 3. Area A = π · 5 · 3 = 15π ≈ 47.124 square units. For the perimeter: h = ((5 − 3)/(5 + 3))² = (0.25)² = 0.0625, so P ≈ π · 8 · (1 + 0.1875 / (10 + √3.8125)) ≈ π · 8 · 1.0156 ≈ 25.527 units. Eccentricity e = √(1 − 9/25) = √0.64 = 0.8. Focal distance c = √(25 − 9) = √16 = 4 units, so the two foci sit on the major axis 4 units either side of the centre. If you change the units to centimetres, every number scales the same way — the formulas are dimensionally consistent.
Frequently asked questions
What is the formula for the area of an ellipse?
A = π·a·b, where a and b are the semi-major and semi-minor axes (the distances from the centre to the edge along the longest and shortest directions). It is a generalisation of the circle area formula A = π·r² — when a = b = r, the ellipse becomes a circle.
How is an ellipse different from an oval?
"Oval" is an informal word for any egg-shaped or stretched-round curve, including ellipses but also other shapes that are not mathematically defined. An ellipse is a precise curve: the set of points where the sum of distances to two fixed foci is constant. So every ellipse is an oval, but not every oval is an ellipse.
Why does the perimeter not have an exact formula?
The perimeter of an ellipse is an elliptic integral, which cannot be written in terms of elementary functions (π, square roots, sines and cosines). Mathematicians have produced increasingly accurate approximations — the calculator uses Ramanujan's second formula, which is accurate to better than 0.00001% for any ellipse. For exact values you would need an infinite series or a numerical integration.
What does eccentricity mean?
Eccentricity e measures how "stretched" the ellipse is. It is zero for a circle (a = b) and approaches one as the ellipse gets longer and thinner. The formula is e = √(1 − b²/a²) where a is the longer axis. Planetary orbits are ellipses with small eccentricities (Earth's is about 0.0167, almost circular).
Where are the foci of an ellipse?
The two foci sit on the major axis, each at distance c = √(a² − b²) from the centre. They have a physical meaning: if you put two pins at the foci and loop a string of length 2a around them, the curve traced by a pencil keeping the string taut is exactly the ellipse. This is why focal distance shows up in everything from planetary orbits (Kepler's first law) to elliptical billiards.
Can I enter the full axes (the diameters) instead of the semi-axes?
No — the inputs are the semi-major and semi-minor axes (half-widths from the centre). If you measured the full long and short diameters, divide each by two before entering. For example, a "10 × 6" ellipse has semi-axes a = 5 and b = 3.