We first need to model the curve (find an equation that represents the curve accurately).

A freely hanging cable takes the form of a catenary. The general form for a catenary is the sum of 2 exponential functions:


The Golden Gate Bridge cable is almost a catenary and almost a parabola, but not quite either (because of the weight of the cables, the suspender ropes and the roadway). For the sake of this discussion, we'll assume it is a catenary.

For convenience, we'll place the origin at the lowest point of the cable.

The required curve (after some guess and check) passing through (-640,152), (0,0) and (640,152) is given by:


Here is the graph of the above equation. We can see it pases through the required points.

Model of Golden Gate Bridge cable

The derivative of our function is


Using the length of a curve formula, with start point x = -640 and end point x = 640, we have:

`int_-640^640 sqrt[1+(640/663(e^[x//1326]+e^[-x//1326])/(2))^2] dx = 1326.956`

So the length of the central span of the main cable is `1327.0\ "m"`. (You can see the answer in Wolfram|Alpha here.)

Of course, the cable continues on both sides of the towers. The total length of each cable is `2,332\ "m"`.

Easy to understand math videos: