You should be able to use similar triangles to solve your problem.
There are several ways to approach it mechanically, but one way is to have a long stick or pole and a measuring tape or yardstick. First find an area of flat ground that has a direct walking path to the base of the tower.
Stick the pole in the ground so it's straight up.
Move your eyeball near the ground and move yourself around until you have lined up the pole with the top of the antenna in your sight line. Mark the spot on the ground directly below your eyeball and measure the height above the ground to your eyeball. Call this height e.
Now measure the height of the top of the pole from the ground (I am assuming you are doing all this on level ground). Subtract e from it and then that is the difference in height from where your eye was and the top of the pole (let's call this measurement h). Next measure the distance from the mark on the ground to the base of the pole. Call this measurement d.
This gives you a triangle to work with: you have the length d and the height h of a right-angle triangle.
The triangle from the mark on the ground to the base of the antenna to the top of the antenna is a similar triangle to this one. So now if you pace off and measure or estimate the distance from your mark on the ground to the base of the antenna, you have a measurement D to compare to d. Let's call H the unknown height of the antenna.
Because these are similar triangles, the ratios of corresponding sides are the same. That means D/d = H/h. You have three of the values, so you can quickly calculate the unknown antenna height.
There are other ways of doing this involving triangulation, shadows, and such, but it really depends on what you are allowed to measure, get close to, and in the case of shadows, your eye protection!
Stephen La Rocque>