# I cannot understand this calculation

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Ok, this is a very stupid math problem but I’m unable to understand it. I'm reading a master thesis called "AN EFFICIENT, HARDWARE-ACCELERATED, LEVELOF- DETAIL RENDERING TECHNIQUE FOR LARGE TERRAINS", in it there's a paragraph with the following statement: "For example, a 100-km2 terrain sampled at 1-meter resolution and represented as a regular grid will contain roughly 20,000,000,000 triangles." How the hell he reached that number? From my calculations I cannot reach that result. 1km2 is 1000000 m2 so 100km2 is 100000000 m2. So at one meter resolution, I can assume that I have 100000000 cells in a grid, because 100000000 m2 is 10000x10000, so I have a square of 10000 per side. So if I have 100000000 cells I should have 2 x 100000000 which is 200000000 triangles (8 zeros) triangles and not 20,000,000,000 triangles (10 zeros) right ?

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(100km)2 = (100,000m)2 = 10,000,000,000m2

10,000,000,000m2 * 2 = 20,000,000,000m2

His answer seems right to me (from skimming over your math, it looks you might be making a conceptual mistake in your area computation).

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Quote:
 Original post by jyk(100km)2 = (100,000m)2 = 10,000,000,000m210,000,000,000m2 * 2 = 20,000,000,000m2His answer seems right to me (from skimming over your math, it looks you might be making a conceptual mistake in your area computation).

First tnks for your answer, it makes sense. But something is still very strange in here if i make a conversion for example in http://www.onlineconversion.com/area.htm from 100 square km to square meters it gives me 100,000,000 square meters not 10,000,000,000. What i'm i missing here ?
(I'm felling very stupid now ;-) as you can imagine )

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It appears that by "100-km2 terrain" the author means a 100 km x 100 km area, rather than an area of 100 km^2. Note that 100 km x 100 km = 10,000 km^2. In other words, the author apparently means (100 km)^2 instead of 100 (km^2). The sentence you quoted is misleading in that it seems to me that the latter interpretation is the most obvious one.

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I'll have to agree with KrazeIke here, I'd interpret it the same way as the OP.

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Quote:
 Original post by KrazeIkeIt appears that by "100-km2 terrain" the author means a 100 km x 100 km area, rather than an area of 100 km^2. Note that 100 km x 100 km = 10,000 km^2. In other words, the author apparently means (100 km)^2 instead of 100 (km^2). The sentence you quoted is misleading in that it seems to me that the latter interpretation is the most obvious one.

Yes I think you are right thank you very much. Sometimes the easiest problem is the hardest one ;-)

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100 square km == 100 * km*km == 100 km^2
100 km, squared ==100*km * 100*km == 100*100 km^2
he just left out the comma

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The author of the paper is confused.

He wrote "a 100-km2 terrain ... will contain roughly 20,000,000,000 triangles". The truth is that it would require 2x108 1-meter triangles, not 2x1010. I say he is confused because followed that with "The Earth has a surface area of more than 300 million km2 ... would require more than 1022 triangles".

The surface of the earth is actually about 500 million km2, but would require only 1x1015 triangles.

I'm disappointed that the advisor didn't catch this. Claiming that the number of triangles required to render the Earth at 1-meter resolution would be the same as the number of stars in the universe is obviously a mistake.

[Edited by - JohnBolton on March 26, 2007 5:29:31 PM]

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Sometimes people use a unit "feet square", where 10 feet square is 100 square feet, 3 feet square is 9 square feet, and so on (ref). It's possible this paper author was using the unit "kilometres square" instead of "square kilometres" or "kilometres squared". Since such a unit is not in common use, this could have produced the resulting confusion. If the original source were available, a better analysis would be possible (if it says km2 then it is simply an error on the part of the paper's author).

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