## Tuesday, August 18, 2015

### 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

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