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How to Draw an Ellipse on the Floor with String and Stakes
Lay out a large floor ellipse using two stakes, a loop of string, and a marked pencil. Includes calculation, stake placement, and a floor-plan check frame.
Free lesson
Four exercises
Worked answers
Object Drawing Lessons
A floor ellipse can be laid out accurately with two fixed stakes and a loop of string. The method relies on the constant-sum property of an ellipse: for any point on the curve, the sum of distances to the two stakes equals the major axis length. This guide gives a calculation box, stake-placement steps, and a check frame so you can verify the curve before committing to paint or tape. All measurements use the same unit, and any numerical examples are hypothetical.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Understand the geometry before marking
An ellipse has a major axis (longest diameter) and a minor axis (shortest diameter). The two stakes sit on the major axis, equally spaced from the centre. The distance from the centre to each stake is called the focal distance. The string loop length equals the major axis plus twice the focal distance, because the loop must reach from one stake to the far end of the major axis and back to the other stake.
The focal distance is found from half the major axis (a) and half the minor axis (b): focal distance = square root of (a squared minus b squared). Keep all values in the same unit, such as centimetres or inches. Label every chosen number as a hypothetical example until you measure your own floor.
- Measure the available floor length and width; choose a major axis and minor axis that fit with clearance.
- Compute a = major axis divided by 2 and b = minor axis divided by 2.
- Compute focal distance = square root of (a squared minus b squared).
- Compute string loop length = major axis + 2 times focal distance.
- Record all values in the calculation box before placing any stake.
Place stakes and form the string loop
Mark the centre of the ellipse on the floor. Measure along the major axis direction and place one stake at the centre plus the focal distance, and the second stake at the centre minus the focal distance. The stakes should be driven firmly so they do not move during tracing. Tie a non-stretching string into a loop of the calculated length, then slip the loop over both stakes.
Insert a pencil or marking tool inside the loop and pull the string taut. The pencil tip traces the ellipse when the string stays taut and the loop remains over both stakes. Keep the string at the same height as the stake tops to avoid length changes.
- Mark the centre point and the major-axis direction line.
- Place stake A at centre plus focal distance; place stake B at centre minus focal distance.
- Tie a string loop equal to the calculated loop length; verify by measuring the loop flat.
- Slip the loop over both stakes and test that it reaches the far major-axis point with slight tension.
- Trace slowly, keeping the string taut and the pencil vertical.
Check the curve against a floor-plan frame
Before painting or taping, verify the traced curve with a simple floor-plan check frame. Mark the four axis endpoints: two on the major axis and two on the minor axis. Measure from each endpoint to each stake and confirm the sum equals the major axis. Also measure the distance between the two minor-axis endpoints; it should equal the minor axis.
A second check is to measure several points around the curve and confirm the sum of distances to the two stakes stays constant within your marking accuracy. If the sum drifts, the string length or stake position has changed. Correct the setup before proceeding.
- Mark the major-axis endpoints and minor-axis endpoints.
- Measure from each endpoint to stake A and stake B; confirm each sum equals the major axis.
- Measure the minor-axis endpoint separation; confirm it equals the minor axis.
- Sample four intermediate curve points and confirm the distance sum is constant.
- Only after all checks pass, apply paint or tape along the traced line.
Worked example
Hypothetical example: lay out an ellipse with major axis 400 cm and minor axis 240 cm. Find the focal distance, stake positions from centre, and string loop length.
Worked answer
a = 200 cm, b = 120 cm. Focal distance = square root of (200 squared minus 120 squared) = square root of (40000 minus 14400) = square root of 25600 = 160 cm. Stakes are 160 cm from the centre along the major axis. Loop length = 400 + 2 times 160 = 720 cm.
Four practice exercises
Exercise 1
Hypothetical example: major axis 300 cm, minor axis 180 cm. Compute focal distance and string loop length.
Show answer and check
a = 150 cm, b = 90 cm. Focal distance = square root of (22500 minus 8100) = square root of 14400 = 120 cm. Loop length = 300 + 240 = 540 cm.
Why: Halve each axis, square, subtract, take the square root for focal distance, then add twice the focal distance to the major axis for the loop.
Check: Verify that 120 squared plus 90 squared equals 150 squared: 14400 + 8100 = 22500, which is 150 squared.
Exercise 2
Hypothetical example: major axis 500 cm, minor axis 300 cm. Where do the stakes go relative to the centre, and what loop length is needed?
Show answer and check
a = 250 cm, b = 150 cm. Focal distance = square root of (62500 minus 22500) = square root of 40000 = 200 cm. Stakes are 200 cm from the centre on the major axis. Loop length = 500 + 400 = 900 cm.
Why: The stakes sit at the focal points, which are the focal distance from the centre, not at the axis ends.
Check: The loop path from one stake to the far major-axis endpoint and back to the other stake is 250 + 400 + 250 = 900 cm, which equals the major axis plus twice the focal distance.
Exercise 3
Hypothetical example: a traced point is 170 cm from stake A and 130 cm from stake B. The major axis is 300 cm. Does the point lie on the ellipse?
Show answer and check
Yes. The sum is 170 + 130 = 300 cm, which equals the major axis.
Why: The defining property is that the sum of distances to the two stakes equals the major axis for every point on the ellipse.
Check: Repeat at two nearby points; if each sum is 300 cm within measurement accuracy, the curve is consistent.
Exercise 4
Hypothetical example: major axis 360 cm, minor axis 216 cm. A helper places stakes 180 cm from the centre. Is that correct?
Show answer and check
No. a = 180 cm, b = 108 cm. Focal distance = square root of (32400 minus 11664) = square root of 20736 = 144 cm. Stakes should be 144 cm from the centre, not 180 cm.
Why: Placing stakes at half the major axis puts them at the axis endpoints, which is a common error; the focal distance is shorter.
Check: Compute 144 squared plus 108 squared = 20736 + 11664 = 32400, which equals 180 squared, confirming the focal distance.
Common mistakes
- Using a stretchy string or cord. Use non-stretching string or mason line; test by hanging a small weight and checking length does not change.
- Placing stakes at the major-axis endpoints instead of the focal points. Compute focal distance from a and b; stakes go at centre plus and minus focal distance, not at the ends.
- Mixing units in the calculation. Convert all measurements to one unit before squaring or adding; label the unit on every value.
- Letting the string ride up or down the stake. Keep the loop at a consistent height, such as at the stake tops, and trace with the pencil vertical.
Self-check rubric
- Calculation box shows a, b, focal distance, and loop length with consistent units.
- Stakes are placed at centre plus and minus focal distance, not at the major-axis endpoints.
- String loop length is verified flat before tracing.
- Curve is checked at four axis endpoints and at least four intermediate points using the distance-sum test.
- Final line is applied only after all checks pass; any deviation is corrected before marking.
Questions and answers
What string length do I need for a given ellipse?
Loop length equals the major axis plus twice the focal distance, where focal distance is the square root of (a squared minus b squared), with a and b being half the major and minor axes.
Can I use this method for a very large floor ellipse?
Yes, provided the stakes are secure and the string does not stretch. For very large spans, use a low-stretch line and check the distance sum at several points.
How do I know the traced curve is a true ellipse?
Measure from multiple curve points to both stakes; the sum of the two distances should equal the major axis within your marking accuracy.
What if the floor is not flat?
The method assumes a flat surface. On a sloped or uneven floor, measure along the surface and keep the string at a consistent height; verify with the distance-sum check.