How to use the ellipse string layout calculator
- Enter the overall length (the long way across) and the overall width (the short way across) of the finished ellipse, and choose a unit.
- Mark the center and the long axis. Place the two pins on the long axis, each at the pin spacing apart, centered on the center.
- Cut a string to the loop length, tie it in a loop (add the knot allowance if the knot uses string), and slip it over both pins.
- Pull the loop tight with a pencil and move the pencil around, keeping the string taut. Use the table of points to check the drawn curve.
Formula
a = length / 2 and b = width / 2
c = sqrt(a^2 - b^2), the distance from the center to each pin
pin spacing = 2c and string loop = 2a + 2c
distance from a pin to the nearer end = a - c
Every point on an ellipse has distances to the two pins that add up to 2a. The string loop passes around both pins and the pencil, so its length is the pin spacing plus 2a. The perimeter uses Ramanujan's second approximation.
An ellipse 72 long and 48 wide
Half the length is a = 36 and half the width is b = 24, so c = sqrt(36^2 - 24^2) = 26.8328. The pins are 53.666 apart, each 9.167 in from the nearer end, and the string loop is 2 x 36 + 2 x 26.8328 = 125.666 long.
The area is pi x 36 x 24 = 2714.336 and the perimeter is about 190.385. The curve passes through 3.6 across and 23.880 high, and through 18 across and 20.785 high. For any point you draw, the distances to the pins add up to 72.
The pins-and-string method
Gardeners lay out oval beds this way, and carpenters use it for tabletops and inlays. It works because an ellipse is exactly the set of points whose distances to two fixed points add up to a constant. A taut string keeps that sum fixed.
The more different the length and width, the farther apart the pins sit. If the two are equal, the shape is a circle, the pins join into one pin at the center, and the loop is twice the radius. For circle measurements, use the Circle Calculator.
The trammel method and accuracy
A trammel is a rod with a pencil at one end and two pins that slide in grooves along the long and short axes. Set the pencil to the first pin at a and to the second at b, and the pencil traces the ellipse as the pins slide. The page gives those distances for you.
Pencil width, string stretch and knots shorten the real ellipse a little, so draw a trial curve and adjust the loop length. For the area of an ellipse and other shapes see the Area Calculator, and for arch templates the Arch Template Calculator.
More layout tools
For other layouts and cutting sizes, try the Dovetail Layout Calculator, Flying Geese Cutting Calculator and Quarter Square Triangle Calculator.
Assumptions and limits
- The ellipse is symmetric, with its long axis along the line through the pins.
- The string does not stretch, and the pencil is a point.
- The width is not greater than the length. Swap them if needed.
- The perimeter is an approximation that is very accurate for ordinary shapes; the pin and string figures are exact.
Frequently asked questions
Why is the string loop longer than the ellipse is long?
The loop is the length of the closed string: it runs from one pin to the pencil and back, and also across the gap between the pins. Its length is the overall length plus the pin spacing.
How do I draw an ellipse that fits a given rectangle?
Use the rectangle's length and width as the overall length and width. The ellipse then touches the middle of each side of the rectangle.
What if the pins are too close together?
A small pin spacing means the length and width are nearly equal. The curve will be close to a circle, and small errors in the loop length matter more, so check the first few points.
Does the knot allowance matter?
Yes. A knot uses some string, which shortens the loop. Enter the extra length your knot takes under More options so the loop ends up the planned length.