Calculate all properties of an ellipse including area, perimeter, eccentricity, and foci distance. Enter any known values for instant results with Ramanujan's perimeter approximation.
An ellipse is a closed curve in a plane where the sum of the distances from any point on the curve to two fixed points (called foci) is constant. It's essentially a stretched circle, and is one of the fundamental shapes in mathematics and nature.
The area of an ellipse is calculated using the formula A = π × a × b, where 'a' is the semi-major axis and 'b' is the semi-minor axis. For example, if a = 8 and b = 5, then Area = π × 8 × 5 = 40π ≈ 125.66 square units. It's similar to a circle's area formula (πr²) but uses both axes.
Unlike circles, ellipses don't have a simple exact perimeter formula. We use Ramanujan's approximation: P ≈ π(3(a+b) - √((3a+b)(a+3b))). This formula is extremely accurate for all ellipses. For very flat ellipses, another approximation is P ≈ π(a + b), but Ramanujan's is much more precise.
Eccentricity (e) measures how “stretched” an ellipse is, calculated as e = √(1 - b²/a²). It ranges from 0 to 1: e = 0 means a perfect circle, while e approaching 1 means a very flat, elongated ellipse. For example, Earth's orbit has e ≈ 0.017 (nearly circular), while Halley's Comet has e ≈ 0.967 (very elongated).
The foci (plural of focus) are two special points inside an ellipse. Any point on the ellipse has the same total distance to both foci. They're located on the major axis, at distance c from the center, where c = √(a² - b²) is the linear eccentricity. The distance between the two foci is 2c. For a circle, the foci merge at the center.
Kepler's First Law states that planets orbit the Sun in elliptical paths with the Sun at one focus. The eccentricity determines how “oval” the orbit is. Most planets have low eccentricity (nearly circular orbits), but comets often have high eccentricity (very elongated). This discovery revolutionized astronomy by showing orbits aren't perfect circles.
The semi-major axis (a) is half the longest diameter of the ellipse - the distance from the center to the farthest edge. The semi-minor axis (b) is half the shortest diameter - the distance from the center to the nearest edge. By convention, a ≥ b. If a = b, the ellipse becomes a circle with radius = a = b.
You can draw an ellipse using two pins (at the foci) and a string. Place pins at the two foci points, loop a string around both pins, and trace with a pencil keeping the string taut. The constant string length ensures every point satisfies the ellipse definition: the sum of distances to both foci is constant. The string length should be 2a.
Linear eccentricity (c) is the distance from the center of the ellipse to either focus. It's calculated as c = √(a² - b²). This value determines where the foci are located along the major axis. The relationship between c, a, and b is fundamental: a² = b² + c². For a circle, c = 0 (foci at center), and for very flat ellipses, c approaches a.
Yes! When the semi-major axis equals the semi-minor axis (a = b), the ellipse becomes a circle. In this case, the eccentricity e = 0, the linear eccentricity c = 0, and both foci merge at the center. A circle is therefore a special case of an ellipse with zero eccentricity. The area formula A = πab becomes A = πr² when a = b = r.
“Perfect for my astronomy project! Being able to calculate orbital eccentricity and visualize the ellipse properties helped me understand Kepler's laws so much better. The Ramanujan perimeter formula is a nice touch!”
“I design elliptical architectural features. This calculator saves me so much time calculating areas, perimeters, and focal points. The export feature is perfect for sharing specs with contractors.”
“Incredibly useful for engineering calculations. The ability to input different parameters and get all ellipse properties instantly is exactly what I needed. The visual charts make presentations easy!”