Mathematics · Geometry · Plane Geometry
Ellipse Area and Perimeter Calculator
Calculates the exact area and approximate perimeter of an ellipse given its semi-major and semi-minor axes.
Calculator
Formula
A is the area of the ellipse. P is the approximate perimeter (circumference) using Ramanujan's second approximation. a is the semi-major axis (the longer half-axis). b is the semi-minor axis (the shorter half-axis). π is the mathematical constant pi (≈ 3.14159).
Source: Ramanujan, S. (1914). Modular equations and approximations to π. Quarterly Journal of Mathematics, 45, 350–372. Area formula from standard Euclidean geometry.
How it works
An ellipse is the set of all points in a plane for which the sum of distances to two fixed points (the foci) is constant. It is fully defined by two parameters: the semi-major axis a (half the longest diameter) and the semi-minor axis b (half the shortest diameter). When a = b, the ellipse becomes a perfect circle with radius a. The ratio between these axes determines how elongated or eccentric the ellipse is.
The area of an ellipse is given exactly by A = πab. This is a direct generalisation of the circle area formula (πr²), where both semi-axes are equal. The perimeter, however, has no exact elementary closed form — it involves an infinite series or elliptic integrals. The most practical and widely used approximation is Ramanujan's second formula: P ≈ π[3(a + b) − √((3a + b)(a + 3b))], which achieves accuracy to within 0.0001% for most practical ellipses. Additional properties such as eccentricity (e = √(1 − b²/a²) for a ≥ b) and the focal distance (c = √(a² − b²)) are also computed, providing a complete geometric characterisation of the ellipse.
Ellipses appear throughout science and engineering: planetary orbits follow elliptical paths (Kepler's First Law), optical and acoustic reflectors exploit elliptical geometry, and elliptical cross-sections appear in structural beams, medical imaging, and stadium design. Accurate area and perimeter calculations are essential in civil engineering for land surveys, in manufacturing for cutting elliptical parts, and in astronomy for computing orbital parameters.
Worked example
Suppose you are designing an elliptical garden bed with a semi-major axis of a = 5 m and a semi-minor axis of b = 3 m.
Step 1 — Calculate the area:
A = π × a × b = π × 5 × 3 = 15π ≈ 47.1239 m²
This tells you how much soil or ground cover you need.
Step 2 — Calculate the perimeter using Ramanujan's formula:
3(a + b) = 3(5 + 3) = 24
(3a + b) = (15 + 3) = 18
(a + 3b) = (5 + 9) = 14
√(18 × 14) = √252 ≈ 15.8745
P ≈ π(24 − 15.8745) = π × 8.1255 ≈ 25.5268 m
This is the length of edging material you need to border the garden.
Step 3 — Calculate eccentricity:
e = √(1 − b²/a²) = √(1 − 9/25) = √(16/25) = 0.8
An eccentricity of 0.8 indicates a moderately elongated ellipse (0 = circle, approaching 1 = very elongated).
Step 4 — Calculate focal distance:
c = √(a² − b²) = √(25 − 9) = √16 = 4.0 m
The two foci are located 4 m from the centre along the major axis.
Limitations & notes
The area formula A = πab is exact and has no limitations for valid positive values of a and b. However, Ramanujan's perimeter approximation, while extremely accurate (error less than 0.0001% for ellipses with eccentricity up to about 0.9), does accumulate small errors for highly eccentric ellipses (eccentricity approaching 1, i.e., very elongated shapes). For such extreme cases, the full elliptic integral formulation or a higher-order series expansion should be used. The calculator requires both a and b to be positive real numbers; entering zero or negative values will produce undefined or nonsensical results. Note also that the semi-major axis must be greater than or equal to the semi-minor axis for the eccentricity and focal distance formulas to apply as stated — the calculator automatically handles which axis is larger when computing these quantities. Units are consistent throughout: if a and b are entered in metres, area is in m² and perimeter in metres.
Frequently asked questions
What is the exact formula for the area of an ellipse?
The exact area of an ellipse is A = πab, where a is the semi-major axis and b is the semi-minor axis. This formula is exact for any ellipse and is a direct generalisation of the circle area formula πr², where both radii are equal.
Why isn't there an exact formula for the perimeter of an ellipse?
The perimeter of an ellipse involves a complete elliptic integral of the second kind, which cannot be expressed as a finite combination of elementary functions (polynomials, exponentials, trigonometric functions). This was proven in the 19th century, which is why mathematicians like Ramanujan developed highly accurate polynomial approximations instead.
How accurate is Ramanujan's perimeter approximation?
Ramanujan's second approximation — P ≈ π[3(a+b) − √((3a+b)(a+3b))] — is accurate to within about 0.0001% for ellipses with eccentricity up to 0.9, and within 0.01% even for very elongated ellipses with eccentricity up to 0.99. It is widely regarded as the best simple closed-form approximation available.
What is eccentricity and what does it tell me about an ellipse?
Eccentricity (e) measures how elongated an ellipse is, ranging from 0 to just below 1. An eccentricity of 0 means the ellipse is a perfect circle. As eccentricity approaches 1, the ellipse becomes increasingly elongated and narrow. Earth's orbit around the Sun, for example, has an eccentricity of about 0.0167 — nearly circular.
What is the difference between the semi-major axis and the semi-minor axis?
The semi-major axis (a) is half the length of the longest diameter of the ellipse, and the semi-minor axis (b) is half the length of the shortest diameter. They are perpendicular to each other and intersect at the centre of the ellipse. If a = b, the ellipse is a circle with radius a.
Last updated: 2025-01-15 · Formula verified against primary sources.