Quick answer
An ellipse with semi-axes a and b has area πab and perimeter approximated by Ramanujan: π[3(a + b) − √((3a + b)(a + 3b))]. With a = b = 5 (a circle) the area is 78.5398 and the perimeter 31.4159.
Enter the semi-major and semi-minor axes. The calculator returns the area and approximate perimeter.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. Circle Calculator: Radius, diameter, circumference and area of a circle from any one of them. Circumference Calculator: Circumference of a circle.
How to use the Ellipse Calculator
- Enter semi-major axis a (above 0).
- Enter semi-minor axis b (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Semi-major axis a | A number | > 0 |
| Semi-minor axis b | A number | > 0 |
Formula and method
Formula
A = πab; P ≈ π[3(a+b) − √((3a+b)(a+3b))] (Ramanujan, first approximation)
IEEE-754 double precision; shown to at most 6 decimal places.
The area is exact. The perimeter has no closed form, so Ramanujan's approximation is used (relative error below 0.5% for moderate eccentricity; exact for a circle).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Semi-major axis a = 5; Semi-minor axis b = 5.
| Result | Value |
|---|---|
| Area | 78.539816 |
| Perimeter (approx.) | 31.415927 |
Example 2
Inputs: Semi-major axis a = 6; Semi-minor axis b = 4.
| Result | Value |
|---|---|
| Area | 75.398224 |
| Perimeter (approx.) | 31.730875 |
The first line is the headline result (area). Below it the ellipse Calculator lists perimeter (approx.), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
No exact perimeter
Unlike the area, the perimeter of an ellipse has no closed-form expression; it requires an elliptic integral. Ramanujan's approximation is accurate to a few parts per million for moderate eccentricity.
Checking with a circle
When a = b the formula collapses to the circle: area π × 25 and perimeter 2π × 5. For a = 5 and b = 3 the area is 47.12 and the perimeter 25.53.
How the result changes with semi-major axis a
Holding every other input at the first worked example, this table changes only semi-major axis a. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Semi-major axis a | Area | Perimeter (approx.) |
|---|---|---|
| 2.5 | 39.269908 | 24.221053 |
| 5 | 78.539816 | 31.415927 |
| 10 | 157.079633 | 48.442105 |
| 20 | 314.159265 | 85.77503 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Semi-major axis a: entering "0" shows "Semi-major axis a must be greater than 0." beside that field and no result.
Choosing between the Ellipse Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
| [Circumference Calculator](/math/circumference/) | i know the and value | Circumference |
| [Area of a Circle Calculator](/math/area-of-a-circle/) | i know the and value | Area |
| [Rectangular Prism Volume Calculator](/math/rectangular-prism-volume/) | length, width and height | Volume |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Perimeter is an approximation
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for semi-major axis a to confirm this.
Frequently asked questions
What are semi-axes?
Half the longest and shortest widths.
How accurate is the perimeter?
Very, for typical shapes.
What is eccentricity?
How stretched the ellipse is.
What does the Ellipse Calculator calculate?
Area and approximate perimeter of an ellipse. It returns Area and Perimeter (approx.).
What formula does the Ellipse Calculator use?
A = πab; P ≈ π[3(a+b) − √((3a+b)(a+3b))] (Ramanujan, first approximation). The area is exact.
What do I need to enter in the ellipse Calculator?
You enter Semi-major axis a and Semi-minor axis b.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Perimeter is an approximation.
What is an example of a ellipse result?
With semi-major axis a 5 and semi-minor axis b 5, the area is 78.539816.
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Circumference Calculator — Circumference of a circle.
- Area of a Circle Calculator — Area of a circle from its radius, diameter or circumference.
- Rectangular Prism Volume Calculator — Volume of a rectangular prism (box).
- Circle Diameter Calculator — Diameter of a circle from radius, circumference or area.
Sources
- Standard Euclidean geometry formulas (e.g. NIST Digital Library of Mathematical Functions §4, CRC Standard Mathematical Tables)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.