Case 1
- Semi-major axis
- 5
- Semi-minor axis
- 3
47.123890
Open with these valuesEllipse Area Calculator
47.123890units²
Result: 47.123890 units²π times the two semi-axes multiplied: πab. Enter the half-lengths, not the full widths. When both are equal it is a circle and the formula becomes πr², which is a good way to check you have entered halves.
The numbers at a glance
Held fixed: Semi-major axis 5.000.
| Semi-minor axis | Result |
|---|---|
| 1.000 | 15.707963 |
| 2.000 | 31.415927 |
| 3.000Your value | 47.123890 |
| 4.000 | 62.831853 |
| 5.000 | 78.539816 |
| 6.000 | 94.247780 |
Worked examples
Case 2
- Semi-major axis
- 10
- Semi-minor axis
- 5
157.079633
Open with these valuesCase 3
- Semi-major axis
- 2.5
- Semi-minor axis
- 2.5
19.634954
Open with these valuesHow it's calculated
A = π × a × b
- StepMeasure the long way across and halve it — that is a.
- StepMeasure the short way across and halve it — that is b.
- ResultRead the area in the square of the unit you entered.
What this number means
Half-lengths go in, not full widths
An ellipse 10 long and 6 wide has semi-axes of 5 and 3, so its area is 15π, about 47.12 square units. Entering 10 and 6 instead gives roughly 188.5 — four times too much.
Equal semi-axes make a circle
Then πab is exactly πr², which is a quick way to check that halves were entered. The rows 1, 1 and 2.5, 2.5 in the table are circles for that reason.
The formula multiplies the axes, it does not average them
Semi-axes of 5 and 3 give 15π, not the 16π an average of 4 would produce. Multiplying is what makes the equal-axis case land exactly on the circle.
The area is exact, the perimeter is not
πab is exact, with no approximation anywhere in it. The perimeter of an ellipse has no exact elementary formula, only approximations such as Ramanujan's.
Commonly misread
An ellipse 10 by 6 has an area of π × 10 × 6.
Those are the full widths. Halve both first: π × 5 × 3 is 15π, about 47.12 square units.
Average the two semi-axes and use πr².
The formula multiplies the semi-axes rather than averaging them. For 5 and 3 the answer is 15π, not 16π.
A circle is not an ellipse, so this calculator cannot do circles.
A circle is the ellipse whose semi-axes are equal. Enter 2.5 and 2.5 and the result is 19.634954, which is exactly π × 2.5².
Reference table
| Semi-axes | Exact | Area |
|---|---|---|
| 1, 1 | π (a circle) | 3.141593 |
| 2.5, 2.5 | 6.25π (a circle) | 19.634954 |
| 4, 2 | 8π | 25.132741 |
| 5, 3 | 15π | 47.123890 |
| 10, 5 | 50π | 157.079633 |
Questions
How do you calculate the area of an ellipse?
Multiply the two semi-axes and then by π. Semi-axes of 5 and 3 give 15π, about 47.12 square units.
Do I enter the full width or the half?
The half. An ellipse 10 long and 6 wide has semi-axes of 5 and 3 — entering 10 and 6 would give four times the true area.
What about the perimeter?
There is no exact elementary formula for it, only approximations such as Ramanujan's. The area, by contrast, is exactly πab with no approximation at all.
Is a circle an ellipse?
Yes, the one where both semi-axes are equal. Then πab becomes πr², which is why the first two rows of the table read as circles.
Sources and last check
Information, not professional advice.