MathChat

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!