A rectangular cuboid has side lengths 5, 7 and 10. Its center is O. The plane P passes through O and is perpendicular to one of the space diagonals. Find the area of its intersection with the cuboid.

Let the cuboid have opposite ends at (0,0,0) and (7,5,10).

One space diagonal is the vector <7,5,10>.  The plane through (3.5,2.5,5) and orthogonal to the vector is 7(x-3.5)+5(y-2.5)+10(z-5)=0
or 7x+5y+10z=87

It will intersect the for long edges at
(0,0,8.7)
(7,0,3.8)
(0,5,6.2)
(7,5,1.3)

Use you favorite method of finding the area of this parallelogram.  (I like the law of cosines but in retrospect should have used cross product.)

To find the exact area as 3.5√174
which seems to agree with Charlie.

Posted by Jer on 2015-05-21 17:07:39