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Ellipse Area Calculator

Calculate the area of an ellipse from its semi-major and semi-minor axes using the formula A = pi a b.

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A = πab

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Oval Garden Bed

An oval garden bed with semi-major axis 4 m and semi-minor axis 2.5 m.

Key values: a = 4 m · b = 2.5 m · A ≈ 31.4 m²

Elliptical Pool

An elliptical pool with semi-major axis 5 m and semi-minor axis 3 m.

Key values: a = 5 m · b = 3 m · A ≈ 47.1 m²

Documentation

Ellipse Area

A=π×a×bA = \pi \times a \times b

where aa is the semi-major axis (half the longer diameter) and bb is the semi-minor axis (half the shorter diameter). When a=b=ra = b = r, this reduces to the circle area πr2\pi r^2.


Eccentricity

The eccentricity measures how “elongated” an ellipse is:

e=1−b2a2,0≤e<1e = \sqrt{1 - \frac{b^2}{a^2}}, \quad 0 \leq e < 1
  • e=0e = 0: perfect circle
  • e→1e \to 1: highly elongated (approaching a line segment)
  • Earth's orbit: e≈0.017e \approx 0.017 (nearly circular)
  • Halley's comet: e≈0.967e \approx 0.967 (very elongated)

Ellipse Perimeter (No Exact Formula)

Unlike the circle, the ellipse has no closed-form perimeter formula. The exact perimeter requires an elliptic integral. Ramanujan's approximation is remarkably accurate:

P≈π[3(a+b)−(3a+b)(a+3b)]P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]

This approximation has a relative error of less than 0.002%0.002\% for all eccentricities.


Frequently Asked Questions

How do I calculate the area of an ellipse?

Use A=π×a×bA = \pi \times a \times b, where aa is the semi-major axis (half the longer diameter) and bb is the semi-minor axis (half the shorter diameter). When a=ba = b, the ellipse is a circle and the formula reduces to πr2\pi r^2.

What is the difference between semi-major and semi-minor axes?

The semi-major axis (aa) is half the longest diameter of the ellipse. The semi-minor axis (bb) is half the shortest diameter. Both are measured from the center to the edge. Always enter the half-lengths, not the full diameters.

What is eccentricity and what does it tell me?

Eccentricity e=1−b2a2e = \sqrt{1 - \frac{b^2}{a^2}} measures how elongated an ellipse is. e=0e = 0 means a perfect circle; ee close to 1 means a very stretched shape. Earth's orbit has e≈0.017e \approx 0.017 (nearly circular).

Is there an exact formula for the perimeter of an ellipse?

No. Unlike the circle, the ellipse has no closed-form perimeter formula — it requires an elliptic integral. Ramanujan's approximation P≈π[3(a+b)−(3a+b)(a+3b)]P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right] is accurate to within 0.002% for all eccentricities.

How does an ellipse relate to a circle?

An ellipse is a circle that has been uniformly stretched along one axis. If you scale a circle of radius rr by factor kk along one axis, you get an ellipse with semi-axes rr and krkr. The area scales by the same factor: πr2\pi r^2 becomes πr×kr=πkr2\pi r \times kr = \pi k r^2.

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