Area
\(A = \pi ab\)
Used when both semi-axes are known.
Ellipse Calculator: Area, Perimeter, Circumference & Eccentricity
Calculate Area and Circumference
Start with the two semi-axes. Use the same length unit for every length input.
Diagram labels update after calculation. The drawing is illustrative, while results use your exact inputs.
Advanced inputs: full axes, area, eccentricity, center
Tip: The main path is \(a\) and \(b\). Advanced values are checked for consistency and used to deduce the same semi-axes when possible.
Recommended Result
Accuracy note
Use Ramanujan II as the standard answer for ellipse circumference/perimeter. It is fast and very accurate for typical geometry, design, landscaping, and measurement work.
- Ramanujan I: quick approximation.
- Ramanujan II: recommended approximation.
- Elliptic-integral series: comparison value using \(C=4aE(e)\).
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Worked Example: a = 5, b = 3
For an ellipse with semi-major axis \(a=5\) and semi-minor axis \(b=3\):
- Area \(= \pi \times 5 \times 3 = 47.124\).
- \(h = ((5-3)^2)/((5+3)^2) = 0.0625\).
- Recommended circumference/perimeter using Ramanujan II \(\approx 25.527\).
Accuracy and Review
Reviewed by: Starlight Tools editorial team, math and engineering calculators.
Last reviewed: June 9, 2026.
Recommended method: Ramanujan II for the displayed circumference/perimeter answer, with Ramanujan I and elliptic-integral series shown for comparison.
References: Srinivasa Ramanujan's 1914 ellipse perimeter approximations and the complete elliptic integral form \(C = 4aE(e)\).
How the Ellipse Area, Perimeter, and Circumference Calculator Works
This calculator helps you find the area and perimeter (circumference) of an ellipse using the values you know. It’s useful for design, engineering, and geometry problems where you need accurate measurements without doing the math by hand.
Think of an ellipse as a stretched circle. The long radius is the semi-major axis \(a\) and the short radius is the semi-minor axis \(b\). The area is straightforward, but the perimeter has no simple exact formula, so we use well-known approximations that are accurate for real-world work.
How to use it
- Enter \(a\) and \(b\), or open advanced inputs for full axes, area, eccentricity, or center coordinates.
- Choose one shared length unit and decimal precision.
- Click Calculate, or let the live form update when enough values are entered.
What the results mean
- Area: computed from \(A=\pi ab\).
- Perimeter/circumference: shown first with the recommended Ramanujan II value, then comparison formulas.
- Eccentricity: tells how “stretched” the ellipse is (0 is a circle, closer to 1 is more elongated).
- Geometry: center, foci, vertices, axis lengths, and the standard equation are shown when semi-axes are available.
All calculations run locally in your browser for speed and privacy.
Ellipse Formulas
Full-axis conversion
\(a = A/2,\quad b = B/2\)
Use this when you know the full major and minor diameters.
Eccentricity
\(e = \sqrt{1-b^2/a^2}\)
Assumes \(a \ge b\). A circle has \(e=0\).
Ramanujan I
\(C \approx \pi[3(a+b)-\sqrt{(3a+b)(a+3b)}]\)
A compact perimeter/circumference estimate.
h value
\(h = \frac{(a-b)^2}{(a+b)^2}\)
Used in Ramanujan II.
Ramanujan II
\(C \approx \pi(a+b)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right)\)
The recommended displayed perimeter/circumference result.
Exact elliptic-integral form
\(C = 4aE(e)\)
\(E(e)\) is the complete elliptic integral of the second kind. This is exact but not elementary.
Standard equation
\(\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}=1\)
Used for the optional center-coordinate outputs.
Ellipse perimeter has no simple elementary exact formula. That is why practical calculators normally recommend a strong approximation, then show an elliptic-integral value for comparison.
Common Ellipse Calculations
Garden beds
Use area to estimate soil, mulch, or ground cover for an oval bed measured by long and short radii.
Area
Tracks and paths
Use circumference/perimeter to estimate edging, fencing, walking distance, or trim around an oval route.
Circumference
Lenses and mirrors
Use eccentricity, foci, and vertices when a drawing needs more than area and perimeter.
Geometry
Arches and openings
Use full-axis conversion when a plan gives total width and height rather than semi-axes.
Diameters
Orbit diagrams
Use center, foci, eccentricity, and the standard equation to label a simplified orbital sketch.
Foci
Ellipse Calculator: FAQs
Is ellipse circumference the same as perimeter?
Yes. For an ellipse, circumference and perimeter both mean the total distance around the curve. This calculator uses both terms because searchers and textbooks use both.
How do I calculate the area of an oval?
For an elliptical oval, multiply pi by the semi-major axis and semi-minor axis: area = \(\pi ab\). If you know full width and height, use \(a = width/2\) and \(b = height/2\) first.
What if I know the major and minor diameters?
Enter the full major axis \(A\) and full minor axis \(B\) in Advanced inputs. The calculator converts them to semi-axes with \(a = A/2\) and \(b = B/2\).
Which perimeter formula should I use?
Use Ramanujan II for most practical ellipse circumference or perimeter calculations. It is very accurate across common ellipse shapes and is shown as the recommended answer.
Why is there no exact simple perimeter formula?
The exact ellipse perimeter is written with a complete elliptic integral, not elementary functions like addition, multiplication, roots, or pi alone. Approximations such as Ramanujan II are used for everyday calculation.
Can I find a missing axis from area?
Yes. If you know area and one semi-axis, the other semi-axis is area divided by \(\pi\) times the known semi-axis. The advanced solver can also use area with eccentricity.
Do all length inputs need the same unit?
Yes. Enter \(a\), \(b\), full axes, and optional center coordinates in the same length unit. Perimeter and circumference use that unit, and area uses the squared unit.
Is my data private?
Yes. Calculations run locally in your browser; nothing is uploaded.