<P> The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth . Excluding the initial 1, this series is geometric with constant ratio r = 4 / 9 . The first term of the geometric series is a = 3 (1 / 9) = 1 / 3, so the sum is </P> <Dl> <Dd> 1 + a 1 − r = 1 + 1 3 1 − 4 9 = 8 5 . (\ displaystyle 1 \, + \, (\ frac (a) (1 - r)) \; = \; 1 \, + \, (\ frac (\ frac (1) (3)) (1 - (\ frac (4) (9)))) \; = \; (\ frac (8) (5)).) </Dd> </Dl> <Dd> 1 + a 1 − r = 1 + 1 3 1 − 4 9 = 8 5 . (\ displaystyle 1 \, + \, (\ frac (a) (1 - r)) \; = \; 1 \, + \, (\ frac (\ frac (1) (3)) (1 - (\ frac (4) (9)))) \; = \; (\ frac (8) (5)).) </Dd> <P> Thus the Koch snowflake has 8 / 5 of the area of the base triangle . </P>

What is the sum of infinite geometric series