they are both types of shapes
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The same as half the side of the square, as the radius of the circle is half its diameter, and the diameter of the circle is equal to the side of the square.
It depends on the diameter of the circle and the width of the square, if they are the same then the answer is no. If you draw yourself a square then inscribe a circle with a radius of half the length of a side of the square, the circle will fit inside the square but the corners of the square will be outside the circle. Thus by inspection the area of the square is larger than the area of the circle.
If the circle touches each edge of the square then its diameter is the same as the side of the square and its circumference is pi times the diameter.
This depends on the circle you're talking about. A theoretical circle and square most certainly could have the same area. If the circle's radius is 1, then the square's length and width would be √π. The problem here is actually in creating such a measurement in a finite number of steps. Because pi is a transcendental number, that is not possible.
A circle or a square.