Calculate area, perimeter, eccentricity, and focal properties from semi-major and semi-minor axes.
Must be ≥ b
Ellipse Diagram
What Is an Ellipse?
An ellipse is a closed, oval-shaped plane curve and one of the four conic sections. It is defined as the set of all points in a plane for which the sum of distances to two fixed points (the foci) is constant and equal to 2a. When the two foci coincide, the ellipse becomes a circle. As the foci move further apart, the ellipse becomes more elongated.
An ellipse is characterized by two axes: the major axis (the longer one, length 2a) and the minor axis (the shorter one, length 2b). The semi-major axis a and semi-minor axis b are half these lengths respectively, measured from the center.
Area Formula: A = πab
The area of an ellipse is elegantly simple:
Area = π × a × b
This formula is a natural generalization of the circle's area. When a = b = r, it gives πr² — the familiar circle formula. The formula is exact, unlike the perimeter approximation. You can verify it via integration: Area = ∫∫ over the ellipse dA = π·a·b.
Perimeter of an Ellipse
Unlike the area, there is no simple exact formula for the perimeter (circumference) of an ellipse. The exact value requires an elliptic integral, a special function that cannot be expressed in terms of elementary operations. The great Indian mathematician Srinivasa Ramanujan discovered an extremely accurate approximation in 1914:
P ≈ π × [3(a+b) − √((3a+b)(a+3b))]
This approximation is accurate to within 1 part in 14 million for most practical ellipses. For near-circular ellipses (small eccentricity) it is even more accurate. A simpler but less accurate approximation is P ≈ 2π√((a²+b²)/2) (within 3% for most ellipses).
Eccentricity Explained
Eccentricity (e) is a number between 0 and 1 that measures how "stretched" an ellipse is. It is calculated as e = √(1 − b²/a²) where a ≥ b. An eccentricity of 0 is a perfect circle; as e approaches 1, the ellipse becomes more and more flattened.
e = 0
Perfect circle (a = b)
e ≈ 0.0167
Earth's orbit
e → 1
Very elongated (parabola at limit)
Ellipses in Nature and Engineering
Planetary Orbits: Kepler's First Law states that planets orbit the Sun in elliptical paths, with the Sun at one focus. Earth's orbit is nearly circular (e ≈ 0.0167).
Stadium Roofs: Many modern sports arenas use elliptical roof designs that distribute structural loads efficiently across the span.
Medical Imaging: Lithotripsy machines use the ellipsoidal reflective property to focus ultrasonic shock waves from one focus to another (the kidney stone).
Optics: Elliptical mirrors reflect light from one focus to the other, used in telescopes, microscopes, and precision optical instruments.
Whispering Galleries: Elliptical room ceilings (like the US Capitol Statuary Hall) create acoustic anomalies — sound whispered at one focus can be heard clearly at the other.
Engineering Drawing: Circles viewed at an angle appear as ellipses. Ellipse templates are standard tools for technical drawings and CAD work.
Comparison with Circle and Other Shapes
Shape
Area
Perimeter
Foci
Circle (a=b=r)
πr²
2Ï€r
One (center)
Ellipse
Ï€ab
Ramanujan approx.
Two (±c, 0)
Rectangle (2a×2b)
4ab
4(a+b)
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Frequently Asked Questions
The area of an ellipse is A = π × a × b, where a is the semi-major axis and b is the semi-minor axis. This is a generalization of the circle area formula: when a = b = r, it gives πr².
The semi-major axis (a) is half the length of the longest diameter. The semi-minor axis (b) is half the length of the shortest diameter, perpendicular to the major axis. By convention a ≥ b > 0.
Yes. When a = b = r, the ellipse equation x²/a² + y²/b² = 1 becomes x²/r² + y²/r² = 1, i.e., x² + y² = r² — a circle. Eccentricity = 0, and both foci merge at the center.
The exact perimeter requires an elliptic integral — a class of integrals that cannot be expressed with elementary functions. Mathematicians have proven this is unavoidable. Ramanujan's approximation is extremely accurate (error < 1 in 14 million) and is widely used in practice.
Eccentricity e = √(1 − b²/a²) measures elongation on a scale of 0 to 1. A circle has e = 0. As e approaches 1, the ellipse becomes a very flat, narrow shape. Earth's orbit (e ≈ 0.0167) is nearly circular; Mars has e ≈ 0.093; Pluto has e ≈ 0.248.
The foci are two special interior points at (±c, 0) where c = √(a²−b²). The defining property: for any point P on the ellipse, distance(P, Fâ‚) + distance(P, Fâ‚‚) = 2a (a constant). This property is exploited in satellite dishes, optics, and lithotripsy.
Ellipses appear in planetary orbits (Kepler's first law), stadium roof designs, elliptical exercise machines, whispering galleries, medical lithotripsy devices, optical mirrors, and gear designs. Any circle viewed at an angle also appears as an ellipse.