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 A Square Problem II (Posted on 2012-05-31)
Given a 1x2x3 cuboid, what is the largest square that can be placed completely inside the cuboid?

 No Solution Yet Submitted by K Sengupta No Rating

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There are two definitions of a cuboid.
(1) For a less restrictive definition, each face of a cuboid is a quadrilateral.
(2) For the more restrictive definition, each face of a cuboid is a rectangle.

For (2), the cuboid is also known as a right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.

As the nomenclature used is simply cuboid, the largest square that can be placed completely inside one would be for the less restrictive cuboid. The side length of largest square that can be placed completely inside this 1x2x3 cuboid would approach 3 as the angles between each of the two opposing vertices approach zero degrees and as the other two angles of the quadrilateral approach 180 degrees.

For (1), the side length of the largest square that can be placed completely inside the 1x2x3 cuboid is sqrt(5).

In both cases, one side of the square shares one edge of the cuboid with its opposite side sharing the opposite edge of the cuboid. The two remaining sides would bisect the 'side' faces of the cuboid.

Edited on June 1, 2012, 9:08 am
 Posted by Dej Mar on 2012-05-31 11:10:42

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