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.
Properties of parallelepiped
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.
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.
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)`
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.