A common application of the area of a circle and the area of a square are problems where a circle is circumscribed about a square or inscribed in a square. Regions between circles and squares problems almost always involve subtracting the two areas; their difficulty stems from dimensions given for one but not both shapes. Related topics include area of sectors and area of circles.
Once you've learned about the areas of circles and squares it's pretty common that on the test or quiz you're going to be having problems where there's kind of an overlap and you're looking for a shaded region that's in between the circle and the square. So here we have an inscribed circle where we've got 4 points of tangency and all we know is the radius. So we're going to do the general case here, if I asked you for the area of a shaded region. So in your mind you have to think well I'm going to be subtracting a couple of things. And to find the area of the shaded region first I'm going to find the area of the square, so notice I'm not actually saying what that area is I'm just kind of setting up a game plan here. So if I find the area of the square and if I take out the area of the circle then what's left is the shaded region. So I'm going to subtract the area of my circle, so it's always a good place to start off these problems where you have shaded regions with a general game plan and then you can substitute in your formulas. Well the area of the square we said is side squared where the side of the square squared, the area of the circle is going to be pi times the radius squared. So what we have to ask ourselves is how is s related to r? Well if I drew in another radius here you can see that, that is going to be the distance of one side. So the side of this squared is 2r so instead of writing s which is a general term we can write that the side of the square must be 2 radii, so we're going to have r squared minus pi r squared. So 2 squared and r squared is 4r squared. So the area of the shaded region is going to be 4 times your radius squared minus pi times your radius squared, which could be rewritten if you factor out the r squared as r squared times 4 minus pi, where that's going to be a number that's going to be less than 1 so you that you're going to have your radius times the number somewhat less than 1. So that's going to be the game plan for solving any problem where you have these shaded regions. Start with the general idea and then substitute in what you're given.