Method & practical interpretation
An ellipse layout you can check.
Start with the finished outline's full dimensions. Keep the centre-based measurements, the focus spacing and the string arrangement separate.
Ellipsepin is a preparation reference for an axis-aligned outline. It calculates one geometric layout from the major and minor diameters you enter. It does not choose a finished size, select a string, account for a particular pencil or compensate for a cutting process. The on-screen diagram gives orientation; the coordinate list and the dimensions provide the numbers to check.
1. Identify the full axes
The full major diameter is the longer end-to-end dimension. The full minor diameter is the shorter end-to-end dimension, measured perpendicular to it through the same centre. Enter those two full lengths. A measurement from the centre to an edge is a semi-axis, so it must be doubled before it can be entered as a diameter.
The calculator sets the centre at coordinate (0, 0), with the major axis running left to right along x. This is a drawing convention, not an instruction to orient your workpiece a particular way. You can rotate the completed reference as a whole; do not independently rotate or shift one focus.
A 1000 × 600 mm outline
Enter major 1000 mm and minor 600 mm. The two half-lengths are a = 500 mm and b = 300 mm. The left and right vertices are (−500, 0) and (500, 0). The bottom and top co-vertices are (0, −300) and (0, 300). These four points are useful checks of the overall dimensions.
2. Read the focus positions
The focus offset is measured from the centre, along the major axis. Ellipsepin calculates c = √((a − b)(a + b)). The product inside the square root is evaluated from the retained exact inputs. This form avoids subtracting two nearly equal rounded squares when the shape is close to a circle. The square root and focus coordinates are then approximate numeric outputs, labelled accordingly.
For the 1000 × 600 mm example, (500 − 300)(500 + 300) = 160000 mm² and c = 400 mm. The foci are at (−400, 0) and (400, 0). Their separation is 800 mm. Confusing the offset with the full spacing would place the pins incorrectly, so both quantities appear separately.
The focus dots in the drawing represent mathematical centres. Their visible diameter is a screen symbol. Neither the screen dot nor the SVG dot specifies a drill diameter, pin thickness or permitted positional error.
3. Use the specified string arrangement
This tool uses one open segment whose two ends are fixed at the two foci. The free working portion runs from one focus to the marking point and back to the other focus. Its two distances add to 2a, which is the full major diameter. In the example, the free working portion is therefore 1000 mm.
A loop around two separated pins is a different physical arrangement. Its route includes an additional part between the pins. Do not take the fixed-end working output and label it a closed-loop perimeter. Ellipsepin does not offer a loop-length mode. Check the arrangement described here against the one you are actually preparing.
The ideal relationship is about the free working segment, not the length of string hidden in knots or wrapped around hardware. If you need extra material for your chosen attachment, enter your own TOTAL extra for both ends. The tool simply adds that number; it does not recommend an amount or predict knot consumption.
Adding an entered allowance
For a 1000 mm working segment, an entered total end allowance of 40 mm gives 1000 + 40 = 1040 mm cut arithmetic. That is 40 mm altogether, not 40 mm at each end. A blank allowance gives an unavailable cut output. Entered zero is different: it explicitly gives the same cut arithmetic as the working segment.
4. Check a second unit example
Enter major 20 in and minor 12 in. The semi-axes are 10 in and 6 in. The focus offset is 8 in, the focus spacing is 16 in and the fixed-end working segment is 20 in. No allowance is implied by those dimensions. Converted to millimetres, the full axes are exactly 508 and 304.8 mm, and the focus offset is 203.2 mm.
Unit changes retain each field's exact physical value. The visible converted text rounds to at most 12 decimal places when needed. It is a display, not a fresh measurement. If you edit that field, its edited text becomes the new value; the other fields retain their own values. Very small dimensions can display as zero after conversion even while their exact retained value remains positive. The conversion status explains this condition. The international inch used here is exactly 25.4 mm.
5. Recognise the circle branch
Equal exact diameters produce a circle. With 100 mm for both axes, the radius is 50 mm and the focus offset and spacing are zero. Both foci coincide at the centre. The diagram shows one focus centre and the text recommends a centre-and-radius construction. It does not invent two separated pin locations for the circle.
Nearly equal axes still produce an ellipse if they differ in the retained exact inputs. There is no hidden tolerance that silently converts it into a circle or swaps a reversed pair. For example, 100 mm and 99.999999999999 mm have a very small positive focus offset. Repeated unit changes retain that distinction until a dimension is edited.
6. Use the reference files carefully
The text report includes exact canonical millimetre values, exact axis fractions in the selected unit and approximate focus coordinates. CSV separates exact and approximate columns. These exports describe your current valid layout only; invalid input clears the diagram and disables export actions. Copying requires browser permission for the clipboard. The text download remains available if clipboard access is refused.
The SVG uses native millimetre width and height. Its viewBox is in the same millimetre coordinate system, centred on the ellipse with padding equal to 3% of the full major diameter at each edge. A 1000 × 600 mm outline therefore has a 1060 × 660 mm outer SVG canvas. The canvas dimensions are not the ellipse dimensions. The graphic includes axis reference lines and focus-centre markers.
Browsers, printers, illustration packages and cutters can apply different sizing defaults. A file declaring millimetres does not guarantee a calibrated print. Check a known dimension after import, disable automatic fit-to-page when appropriate and verify any tiled assembly independently. This is a reference layout rather than a promise of cutting accuracy.
7. Know what the geometry leaves out
The model assumes a flat plane, ideal point foci, an ideal marking point and unstretched string. Real pins and pencils have finite size. String can stretch, shift or wrap, and the attachment can consume a varying length. Those effects are not corrected by the formulas. The tool provides no material tolerance, minimum bend radius or structural result.
Axes must be positive and no larger than 1,000,000 mm equivalent. The minor-to-major ratio must be at least one millionth. Input accepts plain decimals with up to 12 fractional places; exponents and unit suffixes are rejected. These bounds keep this browser implementation predictable. They do not establish practical workshop limits. A very slender valid ellipse may look like a line on screen, so use its numeric dimensions rather than the drawing's apparent thickness.
Major smaller than minor is rejected instead of being automatically exchanged. A negative or missing diameter is also rejected. Correcting the input restores the current layout; Reset example returns to 1000 × 600 mm with 40 mm entered total allowance.
Formula reference
OpenStax, Precalculus 2e: The Ellipse supplies the geometric definition and focus relationship. This site's examples, layout, interface and implementation are original. The calculator does not estimate ellipse perimeter.