Answer: Approximately 7.7 square inches
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Step-by-step explanation:
See the diagram below. We're looking for the area of the shaded region in blue.
This is known as an inscribed circle since the circle is inside the square, touching each side at one point. This circle is as large as possible without spilling outside the square. Think of a balloon that has been inflated as much as possible to fill the room (it touches every wall at a tangent point).
The square has area s^2 = 6^2 = 6*6 = 36 square inches.
The circle has area pi*r^2 = pi*3^2 = 9pi square inches exactly. Note how I cut the diameter (6) in half to get the radius (3).
The blue shaded area is the difference of the square and circle areas, so we have 36 - 9pi as the exact area of the shaded region.
Using your calculator, you should find that 36-9pi = 7.72566611769187 approximately. I'm using my calculator's stored version of pi as opposed to something like pi = 3.14; but you may need to ask your teacher for clarification about what version of pi to use.
When rounding to the nearest tenth, we then get roughly 7.7 square inches as our final answer.