The short answer is
that the area must be equal to the area of the original triangle,
because the Sierpinski gasket (all the “unshaded” regions) has area
zero. See Sierpinski Gasket AreaBut
we can also write a geometric series. Say the initial triangle has area
1. Then the largest shaded triangle had area 1/4. (This is easily shown
using symmetry.) Each of the next smallest triangles occupy 1/16 of the
original area, and there are 3 of them. Then 1/64 of the area, with 9
triangles.At each iteration, there are 3 times
as many triangles but each occupies 1/4 of the previous triangle’s
area. That gives a geometric series with ratio 3/4 for the total area:
