Ellipse Area: worked example
| Semi-major axis (a) | 5 |
|---|---|
| Semi-minor axis (b) | 3 |
| Area | 47.1 |
| Perimeter (approximate) | 25.5 |
Ellipse Area: formula
Enter the semi-major axis, half of the longest diameter, and the semi-minor axis, half of the shortest diameter. Measure across the center, not at the edge. The area is pi times the two semi-axes, and the perimeter is approximated with Ramanujan's formula, since no exact closed form exists.
If both semi-axes are equal, the ellipse is a circle.
Ellipse Area: changing Semi-major axis (a)
| Semi-major axis (a) | Area | Perimeter (approximate) |
|---|---|---|
| 4 | 37.7 | 22.1 |
| 4.5 | 42.4 | 23.8 |
| 5 (example) | 47.1 | 25.5 |
| 5.5 | 51.8 | 27.3 |
| 6 | 56.5 | 29.1 |
Ellipse Area: compare input ranges
| Input | Tested from | To | Area: low input | Area: high input | Swing |
|---|---|---|---|---|---|
| Semi-major axis (a) | 4.5 | 5.5 | 42.4 | 51.8 | 9.4 (20%) |
| Semi-minor axis (b) | 2.7 | 3.3 | 42.4 | 51.8 | 9.4 (20%) |
Ellipse Area: individual input comparisons
| Input | Value | Area | Perimeter (approximate) |
|---|---|---|---|
| Semi-major axis (a) | 4 | 37.7 | 22.1 |
| 4.5 | 42.4 | 23.8 | |
| 5 | 47.1 | 25.5 | |
| 5.5 | 51.8 | 27.3 | |
| 6 | 56.5 | 29.1 | |
| Semi-minor axis (b) | 2.4 | 37.7 | 24 |
| 2.7 | 42.4 | 24.7 | |
| 3 | 47.1 | 25.5 | |
| 3.3 | 51.8 | 26.3 | |
| 3.6 | 56.5 | 27.2 |
Ellipse Area: combined input changes
| Semi-major axis (a) / Semi-minor axis (b) | 2.4 | 3 | 3.6 |
|---|---|---|---|
| 4 | 30.2 | 37.7 | 45.2 |
| 4.5 | 33.9 | 42.4 | 50.9 |
| 5 | 37.7 | 47.1 | 56.5 |
| 5.5 | 41.5 | 51.8 | 62.2 |
| 6 | 45.2 | 56.5 | 67.9 |
Ellipse Area: which assumptions move the result?
Ellipse Area: values and definitions
| Quantity and definition | Value used |
|---|---|
| Semi-major axis (a) — Half of the longest width. | 5 |
| Semi-minor axis (b) — Half of the shortest width. | 3 |
| Area | 47.1 |
| Perimeter (approximate) | 25.5 |
Ellipse Area: half and double Semi-major axis (a)
| Semi-major axis (a) | Area |
|---|---|
| 2.5 | 23.6 |
| 5 | 47.1 |
| 10 | 94.2 |
Common error: Estimating the perimeter as 2 times pi times the average radius. That works for near circles and underestimates a long, thin ellipse noticeably.
The perimeter is an approximation, accurate to a tiny fraction for common shapes and less so for very elongated ellipses. Diameters must be measured through the center. Egg shapes and other ovals are not ellipses.
Rounding of pi has a small effect.
Geometry calculators: Pyramid Volume Calculator, Right Triangle & Pythagorean Calculator, Aquarium Volume Calculator, Annulus Area Calculator, Regular Polygon Calculator, Rhombus Area Calculator.
An ellipse has a clean area formula and a messy perimeter. Area is pi times the two semiaxes, exact for every ellipse with positive semiaxes. The exact perimeter involves a complete elliptic integral of the second kind; Ramanujan’s formula gives an approximation.
A circle is the special case where both half-widths match, and the further apart they drift, the more a circle-based guess for the edge falls short.
Measure half of the widest and half of the narrowest width, both through the center. Using full widths quadruples the area.
Compare with the circle area calculator, the arc length and sector page for circular pieces, and the annulus for rings. The rectangle and regular polygon calculators cover straight-sided shapes, and the unit conversion pages change units.
How is an ellipse related to a circle?
A circle is the special case of an ellipse whose two axes are equal, so the ellipse formulas reduce to the circle formulas when the axes match.
What is the area of an ellipse?
Pi times the semi-major axis times the semi-minor axis. Axes of 5 and 3 give about 47.1.
Why is the perimeter of an ellipse approximate?
There is no simple exact formula, so approximations such as Ramanujan's are used, accurate to a tiny fraction for most ellipses.