Article 6RC91 Golden ellipse

Golden ellipse

by
John
from John D. Cook on (#6RC91)

A golden ellipse is an ellipse whose axes are in golden proportion. That is, the ratio of the major axis length to the minor axis length is the golden ratio = (1 + 5)/2.

Draw a golden ellipse and its inscribed and circumscribed circles. In other words draw the largest circle that can fit inside and the smallest circle outside that contains the ellipse.

goldenellipse1.png

Then the area of the ellipse equals the area of the annulus bounded by the two circles. That is, the area of the green region

goldenellipse2.png

equals the area of the orange region.

goldenellipse3.png

The proof is straight forward. Let a be the semimajor axis andb the semiminor axis, with a = b.

Then the area of the annulus is

(a^2 - b^2) = b^2(^2 - 1).

The area of the ellipse is

ab = b^2.

The result follows because the golden ratio satisfies

^2 - 1 = .

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