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 Appropriate Area (Posted on 2007-03-21)
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.

 No Solution Yet Submitted by Gamer No Rating

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 Solution | Comment 3 of 5 |
`Let the sides of the triangle be`
`    a = 2k+1    b = k^2+k          k>0    c = k^2+k+1     `
`then the semiperimeter is`
`    s = (a+b+c)/2 = (k+1)^2`
`then the area squared is`
`    (Area)^2 = s(s-a)(s-b)(s-c)`
`             = [(k+1)^2][k^2][k+1][k]`
`             = [k(k+1)]^3  `
`This does not list all the triangles, but enough.`
` `

Edited on March 22, 2007, 6:57 pm
 Posted by Bractals on 2007-03-21 21:17:25

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