Urban Trigonometry:



Broadway is a street that runs north-south, but at an angle. It crosses most of the city's north-south avenues and, because the avenues are mostly parallel, forms the same angle at each intersection. As I walked downtown on Fifth one day I realized that the Flatiron Building would provide me with an excellent opportunity:



Problem: What is θ?

On 12/10, I decided to calculate its value.

I set up the following trigonometric situation for the Flatiron Building. All that remained was to find real values for A and B in order to set up a sine ratio.



Rather than measure A and B in meters, feet or millemeters, I chose to use my own paces as units. I did this for a simple logistical reason. Walking around a triangular building multiple times isn't so wierd, but using a tape measure would be awkward in the middle of busy NYC.

I headed up thirteen blocks to the Flatiron Building and started counting paces. The value of A was 32 paces; B turned out to be 78. I checked the lengths again and got the same, give or take a few inches. Man, I'm good!

Just to be sure my step length was consistent, I paced off the third side. I planned to check my consistency with the Pythagorean Theorem -- if all was well, A2 + (the third side)2 would equal B2. The third side checked out at 72. I reviewed my math:

722 + 322 = 6208
782 = 6084
I calculated my experiemental deviation here to be exactly 2%, which is fine. I was glad, because it was rainy outside and I wanted to go back to 35 Fifth.

The grunt work was complete, and finally all I needed to perform was a simple trigonometric calculation. On my walk back to Rubin I thought about the unit circle and and realized that the angle would be less than 30°, but I wanted to get an accurate answer:

sin θ = 32/78

Alternatively, sin-1(32/78) = θ.

My trusty TI-34 came to the rescue...

θ = 24.22°

Whoa! Broadway intersects Fifth Ave (and, presumably, most other avenues) at a much smaller angle than I had anticipated.

Stay tuned; I might get ambitious and head up to Herald Square or Times Square (ha!) to see if this holds steady for all Broadway intersections.