Ellipse Area Calculator
An ellipse is an oval-shaped closed curve that is symmetric about two perpendicular axes. Unlike a circle, which has all points at the same distance from a center point, an ellipse has two focal points, and the sum of the distances from any point on the ellipse to these two focal points is constant. This calculator computes the area and perimeter of an ellipse from the semi-major axis (half the length of the longer diameter) and the semi-minor axis (half the length of the shorter diameter). The area formula is simple: pi times the semi-major axis times the semi-minor axis. The perimeter uses Ramanujan's approximation formula, which provides excellent accuracy for practical applications.
Ellipse formulas
Area = pi * a * b
Perimeter (Ramanujan) = pi * (3(a+b) - sqrt((3a+b)(a+3b)))
Eccentricity = sqrt(1 - (b/a)^2)
Example calculation
For an ellipse with semi-major axis of 10 and semi-minor axis of 6:
- Area = pi * 10 * 6 = 188.50 square units
- Perimeter = pi * (3(10+6) - sqrt((3*10+6)(10+3*6))) = pi * (48 - sqrt(2016)) = 49.62 units
- Eccentricity = sqrt(1 - (6/10)^2) = sqrt(1 - 0.36) = 0.80
Ellipse area calculator: frequently asked questions
What is an ellipse?
An ellipse is a curved, oval shape where the sum of distances from any point on the ellipse to two fixed points (the foci) is constant. A circle is a special case of an ellipse where both foci are at the same point.
What is the formula for the area of an ellipse?
The area of an ellipse is calculated using the formula: Area = pi * a * b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter).
What is the perimeter of an ellipse?
Unlike a circle, an ellipse does not have a simple closed formula for its perimeter. Ramanujan's approximation is commonly used: Perimeter = pi * (3(a+b) - sqrt((3a+b)(a+3b))). This provides a very accurate estimate for most practical purposes.
What are semi-major and semi-minor axes?
The semi-major axis (a) is half the length of the longest diameter of the ellipse, extending from the center to the edge. The semi-minor axis (b) is half the length of the shortest diameter. The ratio of these axes determines how elongated the ellipse is.
How does an ellipse differ from a circle?
A circle has semi-major and semi-minor axes of equal length, so a = b. An ellipse has different lengths for these axes. When a = b, the formula Area = pi * a * b becomes Area = pi * a^2, which is the circle formula.
Official sources
- Wolfram MathWorld: Ellipse.
- ISO 80000-2: Mathematical signs and symbols.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.