November 24, 2008
I don't know why I thought of it, but last week while I was at the National Geographic warehouse sale at the DC Armory I decided that I wanted to know how big DC was from the Capitol to one of the points. I wasn't at home, so I'd have to do the math in my head. And I'm a big enough dork that I did.
According to the Constitution, the Federal District (aka DC) can be, at most, a square 10 miles on a side. The Capitol Building sits at the center of that square. (With Arlington County and Alexandria City having been retroceded to Virginia, the District is no longer a square.) So now we need to find the distance from the center of the square to the corners, knowing the length of the sides.
Cutting from the corners to the center we create a set of 45-45-90 triangles. Since the sides of such a triangle are in the ratio 1:1:sqrt(2) we know that the corner-center length must be 10mi/sqrt(2).
That's a pain to figure out. Let's multiply by sqrt(2)/sqrt(2) and make it 10sqrt(2)mi/2. Much better. Sqrt(2) is about 1.414 so we have 14.14/2, or 7.07 miles.
Now, I don't know how much 0.07 of a mile is. 0.1 of a mile is 528 feet, 0.05mi is 264 feet, and 0.01mi is 52.8. Let's call that 53 for now and come back to it. 0.07 is 0.05 + 0.01 + 0.01, so we'll say 264 + 53 + 53. That's 264 + 106, which is 370. We added 0.2 feet twice, so it's really 369.6 feet.
So how much is 0.6 of a foot? Well, 0.5 is 6 inches and 0.1 is 1.2 inches, so that's 7.2 inches. Let's call it 7 1/4 inches, since that's how rulers are usually marked.
Grand total: 7 miles, 369 feet, 7 1/4 inches. Now how close was I?
Plugging the real value for sqrt(2), or at least as close as my computer can get, into the equation eventually yields 7 miles, 375 feet, 2 7/8 inches. That's a difference of 5' 7 5/8", which is about my height. To use units people actually care about, it's a factor of about 15/100,000, or 0.015%. Not bad for a "quick" guess.
Of course, now that I've said that I should point out that it's impossible to drive exactly 7.07... miles and get to the edge of DC. For starters, your car won't be allowed in the Capitol. Secondly, there are no straight lines to the corners. On the west you have to cross the river onto I-66, which bends. To the east you have to detour around RFK Stadium. To the south the point is in the middle of the Potomac River near the Wilson Bridge. To the north, Capitol Street bends around such that the street that comes closest to running through the point is 16th Street NW, near where a friend of mine used to live in Silver Spring.
But if anyone asks, now you now.
Edit, 4:07 PM: And apparently I was wrong -- the Capitol is not at the center of the District. It appears to be offset just slightly south and quite a bit east. The closest structure to the center seems like the Washington MonumentWhite House, according to the Senate web site. My fault; I just assumed that the center of the grid system was also the center of the square. Still an interesting exercise though.
Edit, 11/26: Before I forget... It turns out the White House isn't dead-center either; the square's center point is just west of the ellipse. 16th Street does make a straight run north, so if any of the points were to be reachable in 7.07 miles, that would have been it. Well, except for the fact that 16th starts a block from the White House.
And looking at Google Maps, I see the White House has been un-obscured -- you can see the roof cleanly now, but it the alleged anti-aircraft guns are up there they've been photoshopped out. Everything within Observatory Circle, not just the Vice President's mansion, are still pixellated. Just in case you were wondering who's really been in charge the last eight years. The Capitol's roof is also plainly visible nowadays, so it's really just Cheney that gets to hide.
