[CZARCAR (OP)]

or round peg in a square hole. Which has the most area for the internal piece?

[imatfaal]

If we take a square which is 2x units to a side

Area of square is   4 x^2 square units

Areas of circle inside is π x^2 square units  (radius is 1x unit)

Area of circle outside is 2π  x^2 square units (radius (√8)x/2 ie √2x units)

take ratios of filled area

circle inside a square is π/4

square inside a circle is 4/(2π)

 π/4      = 0.785398163

 4/(π.2) =  0.636619772

I am sure to have slipped up somewhere

[CliffordK]

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:
πr²

Area of the square.  =  4 x triangle with base&height equal to r
4 x r²/2 = 2r²

Difference is:
πr² - 2r² = (π-2)r² = (3.14-2)r² = 1.14r²

Circle in Square:
Area of circle:
πr²

Area of square = 4 x squares with base&height equal to r
4r²

Difference is:
4r²-πr² = (4-π)r² = (4-3.14)r² = 0.86r²

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:
2r²/πr² = 2/π = 2/3.14 = 0.64

Circle in Square:
πr²/4r² = 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   []

[imatfaal]

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

Phew!

[CZARCAR (OP)]

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:
πr²

Area of the square.  =  4 x triangle with base&height equal to r
4 x r²/2 = 2r²

Difference is:
πr² - 2r² = (π-2)r² = (3.14-2)r² = 1.14r²

Circle in Square:
Area of circle:
πr²

Area of square = 4 x squares with base&height equal to r
4r²

Difference is:
4r²-πr² = (4-π)r² = (4-3.14)r² = 0.86r²

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:
2r²/πr² = 2/π = 2/3.14 = 0.64

Circle in Square:
πr²/4r² = 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   []

great piece i gotta read again but more urgently , i should try to beat a square peg in a round hole instead of a round peg in a square hole?

[imatfaal]

great piece i gotta read again but more urgently , i should try to beat a square peg in a round hole instead of a round peg in a square hole?

in the time you spent to quote and write above - why not read either of the answers given? 

edited to fix quotes

[CliffordK]

in the time you spent to quote and write above - why not read either of the answers given? 

Reading back, it is interesting that we had two different approaches to the same problem.

I used r=radius of circle being the same in both problems...  essentially:

[diagram=630_0]

You, on the other hand used the 2x = side of the square being the same, essentially calculating:

[diagram=631_0]

[imatfaal]

Cliff - I was about to respond, yes but mine is easier; but it isn't really is it?  The only tricky part is that I work out the hypotenuse from two sides and you work out the two sides from the hypotenuse (but you even managed to avoid that - which was nice).  And you went to the trouble of diagrams

[Geezer]

This is a pictorial method of comparing the two ratios.

 [ Invalid Attachment ]

[CZARCAR (OP)]

great piece i gotta read again but more urgently , i should try to beat a square peg in a round hole instead of a round peg in a square hole?

in the time you spent to quote and write above - why not read either of the answers given? 

edited to fix quotes

hard to follow the math & diagrams but simply= both the circle & square's area will be defined by the formation of 2 inner triangles inside both the inner circle & the inner square. The 2 triangles will define the total area of the square. The 2 triangles within the inner circle will define the area of the inner circle but will have extra area that lies outside the 2 triangles.