If you are thinking of a square that precisely fits into a circle.

Or a circle that precisely fits into a square.

[diagram=627_0]

Then you can calculate the areas of each.

Using r as the radius of the circle.

**Square in Circle:**Area of circle:

πr

^{2}Area of the square. = 4 x triangle with base&height equal to r

4 x r

^{2}/2 = 2r

^{2}Difference is:

πr

^{2} - 2r

^{2} = (π-2)r

^{2} = (3.14-2)r

^{2} = 1.14r

^{2}**Circle in Square:**Area of circle:

πr

^{2}Area of square = 4 x squares with base&height equal to r

4r

^{2}Difference is:

4r

^{2}-πr

^{2} = (4-π)r

^{2} = (4-3.14)r

^{2} = 0.86r

^{2}But..

As the previous poster (imatfaal) mentioned, taking differences isn't adequate as the areas are different.

So...

Taking a ratio of the inner to the outer is probably a better way to look at it.

**Square in Circle:**2r

^{2}/πr

^{2} = 2/π = 2/3.14 = 0.64

**Circle in Square:**πr

^{2}/4r

^{2} = π/4 = 3.14/4 = 0.785

So the circle in the square actually wins with more area in the square vs the area in the outer container.

And, it looks like imatfaal and I came up with the same answer [

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