The corners of a shape are called vertices.
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A triangle can't be a square. A triangle only has 3 points, squares have 4.
square pyramid (bottom is a square)-5 points triangular pyramid (bottom is a triangle)-4 points
Half a square is called a rectangle or a triangle.
Such a triangle is an isosceles (and right), but is sometimes called a square triangle (for historic reasons of the definition of square).
That is called a pyramid.