You can calculate the height of a structure too tall to measure directly, such as a flagpole or a building, by geometric or trigonometric methods. In the former case, you compare the shadow of the measured structure to the shadow of a directly measurable object. In the latter case, you view the top of the object through an instrument measuring the angle of view.
How to Calculate Height By Geometry
Plant a stick in the ground on a sunny day and measure its height and the length of its shadow. Denote these measurement "h" and "s" respectively.
Measure the length of the shadow cast by the object being measured. Denote this with the letter "S." A laser distance meter or a surveyor’s scope may be appropriate for this if "S" is too long for measuring tape.
Determine "H," the height of the point on the measured object casting the top of the shadow, by using the proportional relationship between the sides of similar triangles. The stick and its shadow make a triangle similar to the height of the object of interest and the length of its shadow. So, "H/S = h/s." For example, if s=1 meter, h=0.5 meter and S=20 meters, then H=10 meters, the height of the object.
How to Calculate Height By Trigonometry
Determine the angle of the line of sight to the top of the object to be measured. Measure the angle from the ground (as opposed to the angle from vertical). Denote the angle “theta.” A protractor and plumb bob could be made to measure the angle, though a far more accurate measurement can be had from a transit or theodolite -- both surveyor’s tools.
Measure the distance to the object from the same position that you measured the angle. Denote this with the letter "D." Use a laser distance meter or a surveyor’s scope if "D" is too long for measuring tape.
Calculate the height of the object of interest by calculating "D * tan (theta)," where "*" indicates multiplication and "tan" is the tangent of angle theta. For example, if theta is 50 degrees and D is 40 meters, then the height is 40 tan 50 = 47.7 meters, after rounding.
Add the height at which you held the scope to the result of Step 3 for additional accuracy.
Approaches for measuring the height of the top of objects that you cannot measure the horizontal distance to, such as a mountain, include GPS, air pressure and parallax.
One source of inaccuracy in the similar-triangle method is if the measured object tapers. Then the shadow length may not be the full horizontal distance from the top of the shadow to the point under the piece making that part of the shadow. For example, if you are measuring a building that tapers at the top, the top point of the building may be further away horizontally than the shadow length. A mountain is an extreme example of this problem.