Problem 19
Prove that faces, cross sections, and projections of convex polyhedra are convex polygons.
Problem 20
Compute the diagonal of the cube with the edge \(1 \mathrm{~cm}\).
Problem 21
In a cube, which of the two angles is greater: between two diagonals, or between a diagonal and an edge?
Problem 22
Prove that if two diagonals of a rectangular parallelepiped are perpendicular, then its dimensions are congruent to the sides of a right triangle, and vice versa. \mathrm{~cm}$.
Problem 23
Compute the length of a segment if its orthogonal projections to three pairwise perpendicular planes have lengths \(a, b\), and \(c\).
Problem 24
Is a polyhedron necessarily a prism, if two of its faces are congruent polygons with respectively parallel sides, and all other faces are parallelograms? (First allow non-convex polyhedra.)
Problem 26
Prove that in a pyramidal frustum with quadrilateral bases, all diagonals are concurrent, and vice versa, if in a pyramidal frustum, all diagonals are concurrent, then its bases are quadrilateral.
Problem 29
The total surface area of a rectangular parallelepiped is equal to \(1714 \mathrm{~m}^{2}\), and the dimensions of the base are \(25 \mathrm{~m}\) and \(14 \mathrm{~m}\). Compute the lateral surface area and the lateral edge.
Problem 30
In a rectangular parallelepiped with a square base and the altitude \(h\), a cross section through two opposite lateral edges is drawn. Compute the total surface area of the parallelepiped, if the area of the cross section equals \(S\). e.
Problem 31
A regular hexagonal pyramid has the altitude \(h\) and the side of the base \(a\). Compute the lateral edge, apothem, lateral surface area, and total surface area.
