GeomBench

Ellipse Area Calculator

π × a × b — a circle's formula generalised to two radii.

Semi-major axis a

Semi-minor axis b

Area

75.398

Circumference (approx.): 31.731

What you entered

Semi-major a: 6Semi-minor b: 4

1
Area = π × a × b
π × 6 × 4= 75.3982
2
Circumference (Ramanujan's approximation)
π × [3(a+b) − √((3a+b)(a+3b))]= 31.7309

Result

Area: 75.398

An ellipse with semi-axes 6 and 4 has an area of 75.398 and an estimated circumference of 31.731.

Ellipse area and circumference by semi-axes

Area is pi times the two semi-axes multiplied together. Circumference has no exact elementary formula and is approximated.

Semi-major a	Semi-minor b	Area	Circumference
3	2	18.85	15.87
5	3	47.12	25.53
6	4	75.40	31.73
8	5	125.66	41.39
10	6	188.50	51.05
12	8	301.59	63.46

The area formula pi a b is exact and reduces to pi r squared when a and b are equal, since an ellipse with equal axes is a circle. The circumference is different in kind: there is no closed-form expression in elementary functions, so any calculator uses an approximation - Ramanujan's is the usual choice and is accurate to within a tiny fraction of a percent for ordinary shapes. Use semi-axes, measured from the centre, not the full width and height across the ellipse.

A circle is just a special ellipse

When a = b, the ellipse area formula π·a·b collapses to π·r² — the familiar circle area formula. An ellipse is simply a circle stretched along one axis.

Why circumference is only an estimate

Unlike area, an ellipse's circumference has no simple closed-form formula. We use Ramanujan's well-known approximation, which is accurate to within a tiny fraction of a percent for most real ellipses.

Frequently asked questions

How do I calculate the area of an ellipse?

Area = π × a × b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). An ellipse with a=6 and b=4: Area = π × 6 × 4 = 75.4. When a=b, it becomes a circle.

What are semi-major and semi-minor axes?

The semi-major axis (a) is half the longest diameter of the ellipse. The semi-minor axis (b) is half the shortest diameter. They meet at right angles at the center. A = 10 cm long and 6 cm wide means a = 5 cm and b = 3 cm.

How do I estimate the circumference of an ellipse?

There's no exact simple formula. Ramanujan's approximation: C ≈ π × (3(a+b) − √((3a+b)(a+3b))). For a=5, b=3: C ≈ π × (24 − √(18×14)) ≈ π × (24 − 15.87) ≈ 25.53. This is accurate to within 0.04% for most ellipses.