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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

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

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 * 6sqrt(1 + 2 cos(35^o) cos(46^o) cos(60^o)-cos^2(35^o)-cos^2(46^o)-cos^2(60^o)).

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

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

= 120sqrt(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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