How much shorter is Colorado's north border than its south border?

Question

Despite how it appears on the map, Colorado is not rectangular: it's east and west borders are determined by lines of longitude which converge at the north pole (they get closer to each other as they go farther north). Thus, Colorado's northern border must be shorter in length than its southern border. But I have not been able to find anything to say exactly how much shorter (and I must admit, I do not have the math skills to figure it out myself). Does anyone know about this?

Answer 1

That's a great point; Colorado is really a trapezoid! (But even this is only partially accurate; any shape on the surface of a sphere cannot be adequately described in terms of planar figures.) From Wikipedia: "The State of Colorado is defined as the geospherical rectangle that stretches from 37°N to 41°N latitude and from 102°03'W to 109°03'W longitude." So the width is 7 degrees of longitude. That means that the length of the northern and southern boundaries is equal to 7/360 of the circumference of a circle fixed at those latitudes. Given the radius R of Earth (which we will assume to be constant at 6,373 km, even though it varies a bit), the radius at a latitude x would be R*cos(x). The circumference would be 2*pi*R*cos(x), and the length of Colorado's border would be (7/180)*pi*R*cos(x). The northern border is therefore (7/180)*pi*(6373 km)(cos(37°) - cos(41°)) = 34.2 km (or 21.3 miles) shorter than the southern border.

Answer 2

It's easy. Check your maps and find out the longitude of the easstern and western borders. Then find out the latitude of the norhern and southern border. Armed with that information, realize that the length of a small circle of latitude has a total lengthof 25,000 miles times the cosine of the latitude. Since that total length runs around the earth in 360 degrees of longitude. you can calculate how many miles are in one degree of longitude on the northern border and also on the southern border.Subtract those numbers after calculating the total number of miles on each border, and you will yell"AHA AND EUREKA".