How to find lengths of edges of the parallelepiped diagonally

Parallelepiped – a multi-faceted geometric shape, has some interesting properties. Knowledge of these properties helps in solving problems. For example, there exists a certain relationship between its linear and diagonal measurements, which can be used to find lengths of edges of the parallelepiped diagonally.

Instruction

1. The box has one feature that is not characteristic of other figures. Its faces are pairwise parallel and have equal dimension and numerical characteristics, such as area and perimeter. Any pair of these faces could be mistaken for the base, then the rest will be be its side.
2. You can find the edge length of the parallelepiped diagonally, but one of this magnitude is not enough. First, notice what this kind of spatial figures you are given. It can be a right parallelepiped, with right angles and equal dimensions, i.e. cubic In this case is enough to know the length of one diagonal. In all other cases, must be at least one known parameter.
3. Diagonals and length of sides of the parallelepiped associated to a particular ratio. This formula is derived from the spherical law of cosines and represents the equality of the sum of squares of diagonals and the sums of the squares of ribs:
   d12 + d22 + d32 + d42 = 4•A2 + 4•b2 + 4•c2, where a – length b – width c - height.
4. For a cube the formula is simplified:
   4•d2 = 12•A2
   a = d/√3.
5. Example: to find the length of a side of a cube if the diagonal is 5 cm.
   Solution.
   25 = 3•A2
   a = 5/√3.
6. Consider a straight parallelepiped whose lateral edges are perpendicular to the bases, and the bases are parallelograms. Its diagonals are equal and are associated with the lengths of the edges in the following way:
   d12 = A2 + b2 + c2 + 2•a•b•cos α;
   d22 = A2 + b2 +c2 – 2•a•b•cos α, where α is the acute angle between the sides of the base.
7. This formula can be used, if known, for example, one of the sides and angle or these values can be found in other conditions of the problem. The decision is easier when all the angles at the base are straight, then:
   d12 + d22 = 2•A2 + 2•b2 + 2•c2.
8. Example: find the width and height of a cuboid if the width b is greater than the length and 1 cm in height c – 2 times more, and the diagonal d – 3.
   Solution.
   Write down the basic formula of the square diagonal (in a rectangular parallelepiped are equal):
   d2 = A2 + b2 + c2.
9. Express all measurements using a given length a:
   b = a + 1;
   c = a•2;
   d = a•3.
   Substitute into the formula:
   9•A2 = A2 + (a + 1)2 + 4•A2
10. Solve the quadratic equation:
    3•A2 – 2•a – 1 = 0
    Find the lengths of all edges:
    a = 1; b = 2; c = 2.