How to Convert an Ellipse to a Rectangle

"Stretching" an Ellipse to Make a Rectangle

• 1

  Determine the major and minor axes of the ellipse. The major axis is the longest length through the center that touches both sides of the ellipse while the minor axis is the shortest length through the center that touches both sides. You can measure this on a graph, labeling the horizontal dimension as 2*a and the vertical as 2*b. Or you can calculate it by the ellipse equation (x/a)^2 + (y/b)^2 = 1; where 2*a and 2*b are the two axis lengths --- the larger is the major and the smaller is the minor.

• 2

  Define a rectangle with the length of one side given by the major axis and the height of the perpendicular side given by the minor axis. The area of such a rectangle will be length * height = 4*a*b.

• 3

  Draw the rectangle around your ellipse. You will see that the ellipse "just fits" inside the rectangle. If the rectangle were a box that could just hold an elliptical balloon, the balloon could be blown up to squeeze out to the sides of the rectangle.

Constructing a Rectangle with the Area of an Ellipse

• 4

  Calculate the area of the ellipse. Again, start by determining the major and minor axes of the ellipse as in Step 1 of the previous section. The area is given by pi*a*b. Note that this is "a" and "b," the semi-major and semi-minor radii, and not the full axis lengths.

• 5

  Set up the equation for the area of the rectangle, and set it equal to the value of the area of the ellipse. Use the following: Area = Length*Height = pi*a*b. So far, there are an infinite number of rectangles that could be constructed with this area.

• 6

  Assume the ratio of the rectangle sides is the same as the ratio of the major to minor axis of the ellipse. So Length/Height = a/b.

• 7

  Substitute in the area equation to solve for the height. From the equation in the previous step, the Length is (a/b)*Height. Putting this in the area equation, (a/b)*Height*Height = pi*a*b, which leads to Height=sqrt(pi)*b and the Length=sqrt(pi)*a.

• 8

  Draw the rectangle around the same center as your ellipse. You will see that the rectangle is shorter and narrower than the ellipse, but that the corners of the rectangle extend outside the border of the ellipse.