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It was built in a triangle shape. It was built that way because of Broadway. Except for Lower Manhattan, New York City is arranged on a square grid pattern. Broadway ruins this neat little grid pattern by running in a diagonal line north/south through practically all of Manhattan.
Since Broadway bisects some of the rectangular-shaped blocks diagonally, one large rectangular block becomes two smaller triangular blocks. There are more than a few of them dotted around the city, actually.
However, even though Broadway appears to cut diagonally through Manhattan, it actually runs in a straight, true north/south line up and down Manhattan and New York State.
Instead, it's Manhattan that's crooked. All of New York City is built on an eastward slant. Manhattan points northeast, not true north. Broadway starts out in Lower Manhattan and runs north all the way through Manhattan and then almost all the way up through New York State to Canada! It's not called Broadway the whole way, but it's the same street/road. It is the longest street in the world, actually.
So, the Flatiron Building was built in the shape of a triangle because the block that it sits upon is shaped that way.
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