I am not great with calculus and whatnot, but I think this puzzle was within my reach-- Here's what I was able to figure out, and I hope it helps anybody who struggled to understand the math lingo in the other answers: The answer is 45%. This can be seen as a nested stack of triangles, each one 10% narrower and 10% shorter than the last. This is because if all angles are the same, a 10% reduction in height must create a 10% reduction in base width, or else the triangles would no longer be similar. The long form, which I'm sure other answers have written in a more compact mathematical format, is as follows: Following from the top triangle down-  ((.1b*.1h)/2) -- Top triangle, 1st grey area +(((.3b*.3h)/2)-((.2b*.2h)/2)) --2nd grey area +(((.5b*.5h)/2)-((.4b*.4h)/2)) --3rd grey area +(((.7b*.7h)/2)-((.6b*.6h)/2)) --4th grey area +(((.9b*.9h)/2)-((.8b*.8h)/2)) --5th grey area We can strip out the b, h, and even the "/2" portion because these are all in proportion to each other anyway, leaving us with .1² + (.3²-.2²) + (.5²-.4²) - (.7²-.6²) + (.9²-.8²) = .45    

I am not great with calculus and whatnot, but I think this puzzle was within my reach-- Here's what I was able to figure out, and I hope it helps anybody who struggled to understand the math lingo in the other answers:

The answer is

45%.

This can be seen as

.. a nested stack of triangles, each one 10% narrower and 10% shorter than the last. This is because if all angles are the same, a 10% reduction in height must create a 10% reduction in base width, or else the triangles would no longer be similar.

The long form, which I'm sure other answers have written in a more compact mathematical format, is as follows:

Following from the top triangle down- 
((.1b*.1h)/2) -- Top triangle, 1st grey area 
+(((.3b*.3h)/2)-((.2b*.2h)/2)) --2nd grey area 
+(((.5b*.5h)/2)-((.4b*.4h)/2)) --3rd grey area 
+(((.7b*.7h)/2)-((.6b*.6h)/2)) --4th grey area 
+(((.9b*.9h)/2)-((.8b*.8h)/2)) --5th grey area

We can strip out the b, h, and even the "/2" portion because these are all in proportion to each other anyway, leaving us with .1² + (.3²-.2²) + (.5²-.4²) - (.7²-.6²) + (.9²-.8²) = .45 

I am not great with calculus and whatnot, but I think this puzzle was within my reach-- Here's what I was able to figure out, and I hope it helps anybody who struggled to understand the math lingo in the other answers: The answer is 45%. This can be seen as a nested stack of triangles, each one 10% narrower and 10% shorter than the last. This is because if all angles are the same, a 10% reduction in height must create a 10% reduction in base width, or else the triangles would no longer be similar. The long form, which I'm sure other answers have written in a more compact mathematical format, is as follows: Following from the top triangle down- ((.1b*.1h)/2) -- Top triangle, 1st grey area +(((.3b*.3h)/2)-((.2b*.2h)/2)) --2nd grey area +(((.5b*.5h)/2)-((.4b*.4h)/2)) --3rd grey area +(((.7b*.7h)/2)-((.6b*.6h)/2)) --4th grey area +(((.9b*.9h)/2)-((.8b*.8h)/2)) --5th grey area We can strip out the b, h, and even the "/2" portion because these are all in proportion to each other anyway, leaving us with .1² + (.3²-.2²) + (.5²-.4²) - (.7²-.6²) + (.9²-.8²) = .45 [![enter image description here][1]][1] [1]: https://i.sstatic.net/O0aRZ.png

I am not great with calculus and whatnot, but I think this puzzle was within my reach-- Here's what I was able to figure out, and I hope it helps anybody who struggled to understand the math lingo in the other answers: The answer is 45%. This can be seen as a nested stack of triangles, each one 10% narrower and 10% shorter than the last. This is because if all angles are the same, a 10% reduction in height must create a 10% reduction in base width, or else the triangles would no longer be similar. The long form, which I'm sure other answers have written in a more compact mathematical format, is as follows: Following from the top triangle down- ((.1b*.1h)/2) -- Top triangle, 1st grey area +(((.3b*.3h)/2)-((.2b*.2h)/2)) --2nd grey area +(((.5b*.5h)/2)-((.4b*.4h)/2)) --3rd grey area +(((.7b*.7h)/2)-((.6b*.6h)/2)) --4th grey area +(((.9b*.9h)/2)-((.8b*.8h)/2)) --5th grey area We can strip out the b, h, and even the "/2" portion because these are all in proportion to each other anyway, leaving us with .1² + (.3²-.2²) + (.5²-.4²) - (.7²-.6²) + (.9²-.8²) = .45