2. A. Ineffective Ways to Measure

Area of the Koch curve

Replace each small triangle by four still smaller triangles. We see these triangles have base and height 1/3 those of the previous triangles.

Thus these triangles have area 1/9 that of the previous triangles, and so (1/9)2 that of the original triangle.

The areas of these triangles sum to

A2 = ((√3)/12)⋅(16/81) = ((√3)/12)⋅(4/9)2.

By now we should see the general pattern.

Return to Ineffective Ways to Measure.