Center of Gravity of an L-Shaped Sheet and the Golden Ratio

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An L-shaped sheet is constructed by cutting out a square piece from a larger square. The center of gravity (marked as the green point) of the L-shaped sheet shifts as the size of the removed square increases. The center of gravity is outside the L shape if the side of the removed square is greater than the reciprocal of the golden ratio (i.e. ) relative to the side of the larger square.

Contributed by: Sándor Kabai (November 2019)
Open content licensed under CC BY-NC-SA


In this Demonstration, the L shape is divided into two rectangles in two different ways. The center of gravity is then at the intersection of two lines connecting the centers of each pair of rectangles. A proof is given in [3].


[1] Wikipedia. "Locating the Center of Mass." (Nov 8, 2019)

[2] R. Knott. "Two-Dimensional Geometry and the Golden Section or Fascinating Flat Facts about Phi." (Nov 8, 2019)

[3] N. Lord, "Balancing and Golden Rectangles," The Mathematical Gazette, 79(486), 1995 pp. 573–574.


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