why is the area of koch snowflakes finite and the preimetre infinite?

Answer 1

"The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage."

At each stage the number of sides is multiplied by 4, so the sequence of number of sides is 3, 12, 48, 192, ... given by 3*4^n.

At each stage the length of a side is divided by 3, so if the initial triangle has a side of length 1 then the sequence of side lengths is 1, 1/3, 1/9, 1/27,... given by 1/(3^n).

The perimeter = number of sides * length of each side

= 3 * 4^n * 1 / (3^n) = 3 * (4/3)^n

3 * (4/3)^n approaches infinity, as n gets larger. For example, when n = 48, the perimeter is 3*(4/3)^48, about 3 million.

(To say that an expression increases with the number of stages is not enough. The sequence 1/2, 3/4, 7/8, 15/16, ... gets larger and larger but does NOT approach infinity; in fact it approaches 1.)

The area is finite because it lies within a bounded region. Analysis shows that the area is 8/5 times the area of the initial triangle.

Answer 2

With each iteration, the length of the side increases, but the area is bounded. The first picture from the link below will make this easier to understand.