Ellipse Area Calculator
Area of an ellipse from semi-major and semi-minor axes, both unit systems.
What it does
An ellipse is a stretched circle: one radius grows to the semi-major a while the other shrinks to the semi-minor b, and area stretches by the same product. Pi times a times b generalizes pi r squared, which returns exactly when both axes equal the radius.
The tool above takes both semi-axes in centimeters or inches. It returns area with a Ramanujan perimeter approximation and the focal half distance beside it. The default 5 by 3 gives 47.12 cm2 with perimeter 25.53 cm and focal 4 cm.
Precalculus and college geometry sources state the same rule as Area = a·b·π with 2a and 2b as the full major and minor lengths. Worked checks agree: 4 by 3 gives 12π at 37.70, and 6 by 4 gives 24π at 75.40.
Axes mean halves, which is the classic input slip. Full diameters must halve first, since feeding whole lengths quadruples the answer through both factors at once.
How to use
- Pick centimeters or inches.
- Type semi-major a, half the longest diameter.
- Type semi-minor b, half the shortest.
- Read area first, perimeter and focal beside it.
- Check circles: equal axes must match pi r squared.
- Convert mirrors land together through 6.4516.
Formula
- a
- Semi-major axis, half the longest diameter.
- b
- Semi-minor axis, half the shortest diameter.
- c
- Center to focus distance from axis difference.
Imperial inputs convert to centimeters first (x 2.54), then divide square centimeters by 6.4516 for square inches. Perimeter is Ramanujan approximation, exact enough for craft.
Worked example
Example: a 5, b 3
Metric defaults: a 5 cm, b 3 cm.
- Multiply: 5 x 3 = 15.
- Area: 15 x 3.14159 = 47.12 cm2.
- Perimeter: pi x (24 minus root(252)) = 25.53 cm.
- Focal: root(25 minus 9) = 4 cm.
- Imperial check: 1.97 by 1.18 in gives 7.3 in2.
- Tool shows 47.12 cm2, 25.53 cm, 4 cm.
Reference table
| Axes a, b | Exact area | Decimal |
|---|---|---|
| 5, 3 | 15 pi | 47.12 |
| 4, 3 | 12 pi | 37.70 |
| 6, 4 | 24 pi | 75.40 |
| 4, 4 (circle) | 16 pi | 50.27 |
Limits
- Axis aligned ellipses; rotated need transforms.
- Perimeter approximated, never exact.
- Halves assumed; full diameters must halve.
- Units must match across both axes.
Common mistakes
- Feeding full diameters as semi-axes, quadrupling area.
- Using diameter circle math on ovals.
- Mixing cm and inches in one calculation.
- Quoting perimeter as exact.
- Reporting length units for an area.
Try it yourself
Quick checks. Answers sit tucked under each question.
Circle check a = b = 4?
16 pi = 50.27, matching pi r squared.
a 4, b 3?
12 pi = 37.70.
a 6, b 4?
24 pi = 75.40.
Questions
What is the ellipse formula?
Pi times semi-major times semi-minor.
Semi or full axes?
Semi: halve full diameters first.
Default answer?
47.12 cm2 for 5 by 3.
Circle case?
Equal axes return pi r squared exactly.
Perimeter exact?
No closed form exists; Ramanujan approximates within craft tolerance.
Focal distance?
Root of |a squared minus b squared|: 4 cm here.