Compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3), and (10,10,10).

Leonidas Cook

Leonidas Cook

Open question

2022-08-22

Finding volume of convex polyhedron given vertices
I am trying to compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3), and (10,10,10). I am supposed to use a triple integral but am struggling with how to set it up.

Answer & Explanation

Drew Patel

Drew Patel

Beginner2022-08-23Added 9 answers

Step 1
First we can find the equation for the plane that goes through A = ( 1 , 0 , 0 ), B = ( 0 , 2 , 0 ), and C = ( 0 , 0 , 3 ). It is: x + y 2 + z 3 = 1 or 6 x + 3 y + 2 z 6 = 0. Now let O = ( 0 , 0 , 0 ), and D = ( 10 , 10 , 10 ). We now find the volume V 1 of the tetrahedron OABC.
V 1 = 1 6 O A O B O C = 1 2 3 6 = 1. Let d = distance from O to the plane ABC, and S = area of triangle ABC. Then 3 V 1 = S d, and d = | 6 0 + 3 0 + 2 0 6 6 2 + 3 2 + 2 2 | = 6 7 . So 3 1 = S 6 7 .
Step 2
Thus S = 7 2 . Next let k be the distance from D to the plane ABC. Then k = 6 10 + 3 10 + 2 10 6 7 = 104 7 . So let V 2 be the volume of the tetrahedron ABCD. So V 2 = S k 3 = 1 3 7 2 104 7 = 52 3 . So let V be the volume of the polyhedra OABCD, then V = V 1 + V 2 = 1 + 52 3 = 55 3 .

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