11.10. What is the maximum area of the projection of a regular tetrahedron with edge length a onto a plane?
11.10. The projection of a tetrahedron can be a triangle or a quadrilateral. In the first case, it is the projection of one of the faces, so its area does not exceed $\sqrt{3} a^{2} / 4$. In the second case, the diagonals of the quadrilateral are projections of the edges of the tetrahedron, so the area of the projection, being half the product of the lengths of the diagonals by the sine of the angle between them, does not exceed $a^{2} / 2$; equality is achieved when a pair of opposite edges of the tetrahedron are parallel to the given plane. It remains to note that $\sqrt{3} a^{2} / 4 < a^{2} / 2$.