Question

1. what is a better fit, a square peg in a round hole or a round peg in a square hole? i.e. does a circle inside a square waste more or less space than a square inside a circle? hint: pick a radius/side length and calculate areas.

Explanation:

Step1: Let the radius of the circle be \(r\).

Case 1: Circle - in - square

The side length of the square \(s = 2r\). The area of the square \(A_{s1}=(2r)^2 = 4r^{2}\), and the area of the circle \(A_{c}= \pi r^{2}\). The wasted area \(A_{w1}=4r^{2}-\pi r^{2}=(4 - \pi)r^{2}\).

Step2: Case 2: Square - in - circle

The diameter of the circle is the diagonal of the square. If the side length of the square is \(a\), then using the Pythagorean theorem, the diagonal \(d = 2r\), and \(d^{2}=a^{2}+a^{2}\). Since \(d = 2r\), we have \((2r)^{2}=2a^{2}\), so \(a^{2}=2r^{2}\) and the area of the square \(A_{s2}=2r^{2}\). The wasted area \(A_{w2}=\pi r^{2}-2r^{2}=(\pi - 2)r^{2}\).

Step3: Compare the wasted areas

We know that \(4-\pi\approx4 - 3.14=0.86\) and \(\pi - 2\approx3.14 - 2 = 1.14\). Since \(0.86<1.14\), a circle inside a square wastes less space.

Answer:

A round peg in a square hole is a better fit (a circle inside a square wastes less space).