The Math of Triangulation

I think most people have seen a television show or movie where someone has to triangulate a cell phone, a missing hiker, or even an earthquake but what is that exactly?  How do you triangulate?  Whats the math used in it.

Triangulation is defined as a way to use triangles find the source of an earthquake or spacecraft.Although there are several types of triangulation, we are concerned with the type dealing with distance or physical location.

One application is to find the distance of a ship from the shore using trigonometry.  In ancient times, two observers stood on shore a set distance apart.  They then used their instruments to find the angle of the ship and using trig, found the distance.  Maple Soft provided this example.  If you want to check out their interactive activity associated with this, please do. 

I found a lesson on Math Forum which takes the above idea into more depth but instead of using ships, it uses general items placed at certain locations but it takes students through the process of using triangulation to find distances.

Now this example talks about finding out the location of artillery (cannon)  using three towers or microphones using the speed of sound to find the distance.  Its very well written and is actually the answer to a person's question on this topic.  I love the way its explores  a real life situation.

Here is another one on the mathematics behind laser triangulation.  The answer uses the Angle Side Angle theorem to help answer the question.  It is beautifully simple and cool. 

These are just a few instances of triangulation and their math.  Check it out and enjoy.