Unizor - Geometry3D - Prisms - Problems

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Problem 1
Calculate a volume V and the total surface area S of a cube ABCDA'B'C'D', if the length of its main diagonal AC' equals to d.

Answer:
V = d³·√3/9
S = 2d²

Problem 2
Calculate a volume V and the total surface area S of a right regular hexagonal prism with all edges of the same length d.

Answer:
V = 3√3d³/2
S = 3d²(√3+2)

Problem 3
Prove that the area of a square with a side equal to a main diagonal of a right rectangular parallelepiped is greater than or equals to a half of this parallelepiped's total surface, and equality takes place only if this parallelepiped is a cube.

Hint:
Use inequality
x²+y² ≥ 2xy

Problem 4
Prove that the total area of all side faces of any prism (not necessarily the right one) equals to the length of a side edge multiplied by a perimeter of a polygon obtained by cutting the prism by a plane perpendicular to a side edge.

Hint:
Side face is a parallelogram, the area of which equals to a product of a side edge by its altitude.

Problem 5
Given a slanted triangular prism ABCA'B'C' with an isosceles triangle ΔABC as its base, AB=AC.
Side edge AA' forms equal acute angles with base edges AB and AC, that is
∠AA'B = ∠AA'C.
Prove that AA'⊥BC and, consequently, that side face BCC'B' is a rectangle.

Hint:
Prove first the following two simple theorems.
If projection p of line d onto plane γ is perpendicular to some line q on this plane, then the original line d is also perpendicular to line q.
Similarly, if original line d is perpendicular to line q lying on plane γ, then projection p of line d onto this plane γ is also perpendicular to line q.
Symbolically:
p=Projγ(d); q∈γ; p⊥q
⇒ d⊥q
p=Projγ(d); q∈γ; d⊥q
⇒ p⊥q