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What is the true angle of a cheese slope?

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Bricklink lists it as "Slope 30 1 x 1 x 2/3" while LDraw and Peeron list it as "Slope Brick 31 1 x 1 x 2/3". Why do these sources disagree, and what is the real angle?

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up vote 11 down vote accepted

I guess I could send the question to math.se, but I think we can figure that one out ourselves; it's just a bit of trigonometry after all, right?

Let's see: - The large side is 20 ldu (standard brick width). - The small side is 16 ldu (2 plates) - 4 ldu (half a plate), that is 12 ldu (half the height of a brick).

So the angle is tan-1(12/20), which is approximately 30,96°. So it's closer to 31, but gives 30 if truncated down.

As for why the sources differ, I couldn't find documentation on the naming process they both use, but I can easily imagine that someone measuring the angle with a simple protractor could be mistaken by a degree. Another possible explanation would be that the people behind the BrickLink naming chose to round the angle down to simplify things, as a 30° angle feels more natural than a 31° one; while LDraw people may be more used to precise measurements and felt that 31° was closer to the truth. Again, this is all pure speculation, I didn't find naming conventions or guidelines for either.

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does your math account for the fact that the shape isn't technically triangular? –  Mr. Shiny and New 安宇 Jul 9 '13 at 19:37
    
It takes into account the triangular part of the slope only, not the small recess required to allow the stud connection which is present in all slopes, if that's what you're talking about. That's why I substract 4ldu and the height difference between the bottom and top of the triangle is 12ldu. I did neglect the smoothing of the angles, though. –  Joubarc Jul 10 '13 at 6:32
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