Consider a triangle with sides of length 5, 6, 7. If you square the area of that triangle, you get 216, a perfect cube.

Are there other triangles (not geometrically similar to the first triangle) with integral sides whose area squared is a perfect cube? Find one such triangle, or prove no others exist.

Examining right triangles made finding the solution much simpler than other type triangles.  The answer is, yes, there are other triangles with integral sides whose area squared is a perfect cube.

The right triangle of sides 18, 24, and 30 has an area of 216.  Squaring the area (216²) results in 46656, which is 36³.

Another right triangle, with sides 480, 900, and 1020 has an area of 216000.  Squaring the area (216000²) results in 46656000000, which is 3600³.

Edited on March 21, 2007, 2:32 pm

Posted by Dej Mar on 2007-03-21 14:10:15