An area is a part of any surface. Areas are usually separated from everything around them by some kind of boundary, such as a fence or a line drawn around them on a piece of paper.
Area also describes the total size of that area. Different shapes have different formula for calculating how big they are.
A curve is a line that bends to one side or the other along its length. The opposite of curved is straight.
The diameter of a circle is the measurement across the circle at its widest point. The man in the picture has his arms across the diameter of the cut tree.