Question 833510
Each section plus everything above that section base is a triangle.
They are all similar triangles with the side lengths in the proportion
{{{"1 : 2 : 3 : 4 : 5 : 6"}}}.
The length of base and height are also in the same proportion, of course.
So, the areas are in the proportion
{{{1^2:2^2:3^2:4^2:5^2:6^2=1:4:9:16:25:36}}}
The area of the triangle made up of the top two sections is {{{4}}} times the area of the smallest (top) triangle.
The area of the triangle made up of the top three sections is {{{9}}} times the area of the smallest (top) triangle.
The area of the triangle made up of the top four sections is {{{16}}} times the area of the smallest (top) triangle.
The area of the triangle made up of the top five sections is {{{25}}} times the area of the smallest (top) triangle.
The original triangle's area is {{{36}}} times the area of the smallest (top) triangle.
Let's call the area of the smallest (top) triangle {{{x}}} square centimeters.
The area of the triangle made up of the top five sections is {{{25x}}} square centimeters.
The original triangle's area is {{{36x}}} square centimeters.
The difference, {{{36x-25x=11x}}} , is the area, in square centimeters of the bottom section.
Our equation is {{{11x=66}}} --> {{{x=66/11}}} --> {{{x=6}}}.
The area of the original triangle is
{{{36x=36*6=highlight(216)}}} square centimeters.
 
A picture is worth a thousand words:
{{{drawing(300,300,-1,7,-1,7,
triangle(0,0,6,0,4,6),
red(line(-0.5,1,6.5,1)),red(line(-0.5,2,6.5,2)),
red(line(-0.5,3,6.5,3)),red(line(-0.5,4,6.5,4)),
red(line(-0.5,5,6.5,5)),green(line(1,0,4.33,5)),
green(line(2,0,4.67,4)),green(line(3,0,5,3)),
green(line(4,0,5.33,2)),green(line(5,0,5.67,1)),
green(line(1,0,0.67,1)),green(line(2,0,1.33,2)),
green(line(4,0,2.67,4)),green(line(3,0,2,3)),
green(line(5,0,3.33,5))
)}}} I added green lines to split the whole thing into {{{36}}} triangles that are congruent with (the same as) the smallest (top) triangle.
Do you see hoe there are {{{36}}} small triangles in all and {{{11}}} of them in the bottom section?