Area of the Koch Snowflake

The first observation is that the area of an equilateral triangle with side length a is

as we can determine from the following picture

The area of the initial triangle T is therefore .

Area of first iteration:

Area of second iteration:

Area of third iteration:

By now the pattern should be clear. At the kth iteration we add 3*4^(k-1) additional triangles of area . This means we add a total area of

to the area S(k-1) to get the area of S(k). Hence after n iterations we get the area of S(n) to be

The sum inside the parentheses is the partial sum of a geometric series with ratio r = 4/9. Therefore the sum converges as n goes to infinity, so we finally obtain