Ellipse Circumference Calculator
Ramanujan perimeter and area from semi axes.
What it does
Circles get 2 pi r while ellipses get no such luck: the true perimeter is an elliptic integral with no elementary closed form. Craftsmen, runners, and gardeners still need a number, so a great approximation does the job.
The tool above takes the two semi axes in either order. It returns the Ramanujan second approximation with the exact area pi a b beside it, in inches or centimeters.
The defaults set 5 by 3: circumference 25.53 with area 47.12 sq in. The metric mirror (12.7 cm, 7.62 cm) gives 64.84 cm with 304.02 sq cm, and a 5 by 5 circle checks at 31.42.
The approximation matches the published Ramanujan second form with the exact 4aE(e) integral behind it. Peer reviewed ellipse work states both side by side, with NIST digital library definitions covering the integral half.
How to use
- Pick imperial (in) or metric (cm).
- Enter the two semi axes, any order.
- Halve full widths first if you measured across.
- Read circumference first.
- Read area beside it.
- Trust the circle check when axes match.
Formula
- a, b
- Semi major and semi minor; tool sorts them.
- h
- Shape number from 0 (circle) to 1 (flat).
- C
- Ramanujan second approximation of perimeter.
- E(e)
- Complete elliptic integral behind the exact value 4aE(e).
Equal axes send h to zero and return the circle exactly. Very flat ellipses carry the largest approximation error, still tiny for ordinary shapes, while degenerate zero axes are rejected outright.
Worked example
Example: semi axes 5 by 3
Defaults: imperial, axes 5 in and 3 in.
- h: ((5-3)/8)^2 = 0.0625.
- Root: sqrt(4 - 0.1875) = 1.9526.
- Factor: 1 + 0.1875/11.9526 = 1.01569.
- C: pi x 8 x 1.01569 = 25.53.
- Area: pi x 5 x 3 = 47.12 sq in.
- Tool shows 25.53 in, 47.12 sq in. Metric gives 64.84 cm, 304.02 sq cm.
Reference table
| Axes | Perimeter | Area |
|---|---|---|
| 5 by 5 (circle) | 31.42 in | 78.54 sq in |
| 5 by 3 | 25.53 in | 47.12 sq in |
| 10 by 2 | 42.02 in | 62.83 sq in |
Limits
- Approximation, not the exact integral.
- Thinnest ellipses carry the most error.
- Zero or negative axes are rejected.
- Measured ovals are rarely true ellipses.
Common mistakes
- Entering full diameters as semi axes.
- Mixing inches and centimeters.
- Expecting an exact elementary formula.
- Using perimeter where area belongs.
- Trusting tape measures past their slack.
Try it yourself
Quick checks. Answers sit tucked under each question.
Circle 5 by 5 gives what?
31.42 in with 78.54 sq in.
Axes entered 3 by 5 instead?
Same 25.53 in; order never matters.
Long thin 10 by 2 gives what?
42.02 in with 62.83 sq in.
Questions
Is there an exact ellipse perimeter formula?
Only as 4aE(e) with the elliptic integral; no elementary closed form exists.
How accurate is Ramanujan II?
Within about 0.04% across ordinary shapes, exact for circles.
Semi axis or full width?
Semi: half the full width. Halve diameters before entering.
Does order matter?
No; the tool sorts the axes before the math.
Why two outputs?
Trim and edging need perimeter; paint and sod need area.
Garden ellipse beds?
Stake the two axes, run this page, buy edging to the perimeter.