It depends in which shape you want to inscribe it e.g. circle, triangle, hexagon etc. If you provide more information, someone should be able to tell you.
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A circle with radius 15mm will fit in a 30mm square. Find the intersection of the square's diagonals, that is the center of the circle.
Use a compass to inscribe polygons in a circle.
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.
A pair of compasses and straight edge will suffice.
Find the total area of the square: length times height. Next, find the total area of the circle: Pi times radius to the second power, or Pi(r squared). If you are doing this by hand, 3.14 is usually acceptable for Pi. Once you have the are of both the square and the circle (the area of the circle should be smaller than that of the square), subtract the area of the circle from the area of the square. The difference is the area of those extra corners of the square that the circle does not occupy. It is actually quite simple. This demonstrates the danger of thinking in words rather than pictures.