Edges are found in 3-dimensional objects. Then vertices are the corners and edges are the lines formed where two faces meet. In this case, sides usually refers to the faces.
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A shape with four sides has more vertices because there are more points where edges (straight lines) meet than a 3-sided shape.
A rectangle has 4 sides and 4 vertices
An octagon has 8 sides with 8 vertices and one flat face
The edges of a square are also known as the sides. These are the straight lines which form the boundaries of a square. The vertices are the four corners or angles where two sides meet.
A triangle has 3 sides (edges) and three vertices (corners)