Answer to problem about the circular and square pegs and holes.

The “fit” for each situation is the following ratio:

(Area of Inner Figure) ÷ (Area of Outer Figure)

For the square peg in a round hole —

Call the radius of the circle r.

Then the diagonal of square “peg” = 2r

Notice that by slicing the square along its diagonal,

we get a 45-45-90 triangle, with the diagonal being

the hypotenuse and the sides being the two equal legs.

Using the proportions in a 45-45-90 triangle,

side of square peg = r times the square root of 2

Multiplying this side of the square by itself gives

us the area of the square, which comes out as:

2 times the radius squared

This being the case,

Area of square is: 2 times radius squared, and

Area of circle is: Pi times radius squared, and so …

Cancelling the value of the radius squared, we get:

Ratio of (Area of square) to (Area of circle) is:

2÷Pi = 0.6366

For the round peg in a square hole —

Call radius of the circle r.

And since the diameter of the circle is the same length as

the side of the square, the side of the square = 2r

Multiplying the side of the square by itself to get the

area of the square, we find that the area of the square

is given by: 4 times radius squared.

This being the case,

Area of circle is: Pi times radius squared

Area of square is: 4 times radius squared, and so …

Ratio of (Area of circle) to (Area of square) is therefore:

Pi ÷ 4 = 0.7854

Of the two ratios, the ratio of the circular peg in a square hole

is greater than that of the square peg in a circular hole.

Therefore we can say that the circular peg in a square hole

provides a better fit than a square peg in a circular hole.

And that is the answer!

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