Problem: Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection.
Source: Arnold's Trivium.
2026-04-14 03:26:46.1776137206
Expected value of randomly distributed projection
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The area of the projection is the area that you see when looking at the cube. You see three faces, and the area under which a face appears is its actual area, $1$, times the absolute value of one of the coordinates of the uniformly random unit direction vector. Those coordinates are uniformly distributed over $[-1,1]$ (see Generate a random direction within a cone), so the mean absolute value is $1/2$, so by linearity of expectation the expected projected area is $3/2$.