In response to the question of how one measures the height of a mountain, another user stated that an instrument called a transit is used. Using geometry and a transit, somebody can find the height of a mountain in the same way she would find the length of a side of a triangle using the triangle's other side lengths and angle measures. But how do you determine the first side lengths and angle measures? To measure the height of a mountain, that has to be one huge triangle!

Yes - you basically use geometry on some very big triangles. For example:

1. From a known height, measure the angle to the top of the mountain with your transit. Call the angle you measure above the horizontal \theta. If you know the distance from your position to the mountain top (d), then the elevation of the moutain top above your current point is simply d*sin(\theta). The only problem is how do you determine the distance d?

2. Measure the angle to the mountain top as above, then move a known distance closer to the mountain and remeasure the angle. To keep it simple, let's assume that the elevations of the two points you do the measuring from are the same. If the first angle is \theta , and you move a distance x closer to the mountain and measure the second angle \alpha, then the original distance of the mountain top from the first point was:

d = x* \frac {sin(\alpha)} {sin(\alpha-\theta)}


3. Another way to measure the distance to the mountain is to measure its bearing from a first point, move laterally a known distance to a new point and take a new bearing from there. If you move laterally a distance x, and measure a change in bearing of \theta, then the original distance d to the mountain was:

d = x/tan(\theta)


Basically if you can take an elevation reading and bearing from any two known points you can determine the height of the mountain. By the way, this is what George Everest did when he surveyed the Himalaya Mountains back in the late mid-1800's - he was never closer than 50 miles or so from what later became known as Mt. Everest, and had no idea that he had surveyed the tallest mountain in the world until he "crunched the numbers" much like this.

For anyone wanting to get an appreciation of how surveying was done before modern surveying instruments were available (never mind GPS) you might find The Measure of All Things ,by Adler, interesting. It was not for the faint of heart.

I'm a surveyor, and I know that the simplest answer to give to you is, that the easiest way for you to calculate mountain heights is to use a topographic map and subtract the bottoms elevation ( by using the contour lines ) , from the top's. I could give you trigonometric formulae, but I know you are probably not going to apply it in the field.