Your typical American border is the straight line, as demonstrated by the US-Canadian border that follows the 49th parallel for approximately 1.245 miles (2.000 km), longer than any other linear boundary; and by Utah Phillips’ observation that “out west the states are square”.
A delightful exception to the straight border is the circular demarcation between Pennsylvania and Delaware, dubbed the Twelve Mile Circle. This is the only US boundary that’s a true arc… Unless you consider the 49th parallel border and all the other boundaries based on latitude (which are therefore centred around the North Pole) as arcs too.
As the name implies, the circle has a radius of exactly 12 miles, centred on the cupola of the New Castle courthouse. The centre of the circle has been fixed on that cupola since 1750, but the Twelve Mile Circle is older than that, dating back to the original deed of Delaware by the Duke of York to William Penn, on August 24th of 1682:
“All that the Towne of Newcastle otherwise called Delaware and All that Tract of Land lying within the Compass or Circle of Twelve Miles about the same scituate lying and being upon the River Delaware in America And all Islands in the same River Delaware and the said River and Soyle thereof lying North of the Southermost part of the said Circle of Twelve Miles about the said Towne.”
This paragraph has caused another demarcatory anomaly in the Delaware River, the border between Delaware and New Jersey. In most rivers that divide two political entities, the boundary is drawn right down the middle of the stream. Yet in this river, the Twelve Mile Circle continues into the river up unto the New Jersey shoreline. And only there does the state of Delaware stop, claiming the entire river and hemming in New Jersey.
New Jersey has challenged this demarcation up to the Supreme Court (in 1934 and 1935), which refused to rule and instead reprimanded the states for even fighting about this. And yet, as recently as 2006, a study was commissioned on the border dispute. Legislators of both states have made aggressive noises about each others’ claims, Delaware symbolically calling upon the National Guard to defend state shores and New Jerseyites obliquely mentioning the battleship named after their state, moored just upriver…
This image taken from Wikipedia, here.