H = 6?

The height of this gray rectangle is larger than the 6, right? If you rotate the purple line to the left around it's lower left origing then the top of the purple line will start to show a gap below the white line that moves to the top left corner. Hence the area of the rectangle > 60, hence the triangle > 30,

Any good teacher would point out this sanity check

But the width of the gray rectangle is SMALLER than the 10, right? If you rotate the white line to the top around it's upper left origin then the right edge of the white line will start to extend past the rectangle and the top right left corner. Hence the area of the rectangle < 60, hence the triangle < 30,

So how do the two line-to-side ratios differ and does the effect of the purple line outweigh the effect of the while line?

The answer is that they exactly balance each other. The height of the rectangle is 6/cos(theta) and the width of the rectangle is 10*cos(theta) where theta is the narrow angle found in both the large and small triangles.

Thus the rectangle is exactly 6*10 and the triangle is exactly 30.