Problem 32
Compute the total surface area of the tetrahedron all of whose edges have the same length \(a\).
Problem 34
Prove that if all lateral edges of a pyramid form congruent angles with the base, then the base can be inscribed into a circle.
Problem 35
Prove that if all lateral faces of a pyramid form congruent angles with the base, then the base can be circumscribed about a circle.
Problem 36
A regular hexagonal pyramid, which has the altitude \(15 \mathrm{~cm}\) and the side of the base \(5 \mathrm{~cm}\), is intersected by a plane parallel to the base. Compute the distance from this plane to the vertex, if the area of the cross section is equal to \(\frac{2}{3} \sqrt{3} \mathrm{~cm}^{2}\).
Problem 37
The altitude of a regular pyramidal frustum with a square base is \(h\), and the areas of the bases are \(a\) and \(b\). Find the total surface area of the frustum.
Problem 38
The bases of a pyramidal frustum have areas 36 and 16 . The frustum is intersected by a plane parallel to the bases and bisecting the altitude. Compute the area of the cross section.
Problem 39
Through each edge of a cube, draw outside the cube the plane making \(45^{\circ}\) angles with the adjacent faces. Compute the surface area of the polyhedron bounded by these planes, assuming that the edges of the cube have length \(a\). Is this polyhedron a prism?
Problem 40
Prove that if all altitudes of a tetrahedron are concurrent, then each pair of opposite edges are perpendicular, and vice versa.
Problem 41
Prove that if one of the altitudes of a tetrahedron passes through the orthocenter of the opposite face, then the same property holds true for the other three altitudes.
Problem 51
Compute the volume of a regular triangular prism whose lateral edge is \(l\) and the side of the base is \(a\).
