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Parallelepiped

The parallelepiped is a three-dimensional geometry shape. It is constructing with six parallelograms. It is related with parallelogram just like cube with square in Euclidean geometry.

 

Explanation for Parallelepiped

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The following definitions help the students to understand that shape.
  • The shape contains parallelogram and polyhedron with six faces.
  • It is constructed with 3 pairs of parallel faces that is hexahedron.
  • It is a prism with parallelogram base.

Properties of parallelepiped

  • The base of prism is viewed with three pairs of parallel faces.
  • This shape is formed from linear transformations of a cube.
  • Another name of parallelepiped is zonohedron.
  • It is closed with four rectangular faces and two rhombic faces.

Volume of parallelepiped:

 The formula for volume of parallelepiped is

V = `abcsqrt(1 + 2 cos(alpha) cos(beta) cos(gamma)- cos^2(alpha)-cos^2(beta)- cos^2(gamma))`

Here the a, b and c are length of edges and `alpha, beta and gamma` are internal angles.

Examples

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Lets see solved problem on parallelepiped

Problem: Find the parallelepiped volume with lengths a = 5, b = 4 and c = 6, the internal angles are `alpha` = 35o, `beta` = 46o and `gamma` = 60o.

Solution:

Given values are a = 5, b = 4 and c = 6, the internal angles are `alpha ` = 35, `beta` = 46 and `gamma ` = 60.

The volume of parallelepiped is `abcsqrt(1 + 2 cos(alpha) cos(beta) cos(gamma)- cos^2(alpha)-cos^2(beta)-cos^2(gamma))` .

V = 5 * 4 * 6`sqrt(1 + 2 cos(35^o) cos(46^o) cos(60^o)-cos^2(35^o)-cos^2(46^o)-cos^2(60^o)).`

   = 120`sqrt(1 + 2(0.819)(0.694)(0.5)-(0.670)-(0.481)-(0.25))`

   = 120`sqrt(1 + 0.568-0.670-0.481-0.25)`

   = 120`sqrt(0.167)`

   = 49.03

Therefore, the volume of parallelepiped is 49.03.

Exercise problem for parallelepiped:

1. Find out the volume of parallelepiped with a = 10, b = 15 and c = 18, the internal angles are `alpha ` = 90, `beta` = 50 and `gamma` = 47.

Solution: The volume is 977.23.


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