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There is more than one possibility.

If by a diamond you mean a square tilted up onto a corner, and you wnt a 3D shape something like that, probably the best answer is an octahedron, which has 8 triangles as faces; see http://en.wikipedia.org/wiki/Octahedron.

You might also go for the triangular dipyramid, which has 6 triangular faces and is made by sticking two triangular pyramids together;
see http://polyhedra.org/poly/show/56/triangular_dipyramid - you can rotate the image to get a better idea of what it looks like. There are other dipyramids: the square dipyramid is the same as the octahedron; there is also a pentagonal dipyramid and more.

It is better to actually make some of these by cutting equilateral triangles out of paper and sticking them together.

If you mean what shape is a real diamond in a piece of jewellery, that is a specialised topic, which I don't know about. There are some images of different diamond cuts at
http://en.wikipedia.org/wiki/Diamond_cut.
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Q: What does a 3D diamond shape look like?
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A parallellopiped


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