Area of an Ellipse

Calculates the area of an ellipse as π × semi-major axis × semi-minor axis. When the two radii are equal this reduces to the area of a circle, πr². The perimeter is given by Ramanujan's approximation.

Explanation

An ellipse is a circle stretched in one direction. The distance from the centre to the farthest point is the semi-major axis $a$ , and the distance to the nearest point is the semi-minor axis $b$ . The area is just their product, times $\pi$ .

S = \pi a b

$S$ — the area
$a$ — the semi-major axis, the longer radius
$b$ — the semi-minor axis, the shorter radius

When $a = b$ the ellipse is a circle and the formula reduces to $\pi r^2$ .

Example

With the defaults, $a = 5$ and $b = 3$ :

S = \pi \times 5 \times 3 = 15\pi \approx 47.1239

The area is about 47.1239. For the perimeter the calculator uses Ramanujan's approximation

L \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right]

Here $3(a + b) = 24$ and $(3a + b)(a + 3b) = 18 \times 14 = 252$ , so $L \approx \pi (24 - \sqrt{252}) \approx 25.5270$ .

Watch out

The area formula is exact, but there is no elementary closed form for the perimeter of an ellipse. The true perimeter is an elliptic integral, and it cannot be written with finitely many arithmetic operations and square roots. The figure shown here comes from Ramanujan's approximation, whose error is minute unless $a$ and $b$ are wildly different. Both axes must be positive, and swapping them leaves the area and the perimeter unchanged.