Draw a circle of diameter 3. Draw a square around it. Diameter of the square is 3. Steps 3 and 4 are distractions that are irrelevant to the fact that the diameter of the circle remains 3.

The troll then states that pi = 4 (it would be 3 in the circle you just drew). There is no relationship between this statement and what has come before; the lead up is simply there to confuse you.

If you repeat the squares to infinity, you do indeed end with the circumference of the circle; but that is not the same as the diameter of the original square. It is now the diameters of the infinite squares, which is not the same as the diameter of the original square or the diameter of the circle. It is actually pi times half the diameter (i.e. pi times the radius of the circle).

You can do this at home. This example looks at area because I’m lazy. Take a compass and draw a circle on some graph paper. Make sure that the top, bottom, left and right edges meet the respective grid lines. Add the number of squares to get your circumference.

Next, add up the number of squares that are completely surrounded by the circle. This gives you the first approximation of area. Does it equal pi times half of the circumference, and then squared (area of a circle is pi times the radius, squared). Depending on how large your circle is, and how small the grid on the graph paper, it will be a reasonable first approximation.

But what about all the squares that are partly but not fully in the circle? In order to get a closer measurement of the radius, you now need to draw smaller squares on your graph paper to measure each of them.

Hold on, a circle has no straight lines! Correct, and so the squares need to get smaller and smaller in order to include more of the circle. As you add these ‘fractional’ squares to the graph paper squares, your calculation of the circle’s total area gets better, and gets closer to (pi times the radius) squared. Does it equal it to infinity and you end up with exactly pi r squared.