You can easily measure the height of a flagpole without having to climb it by using the rule of similar triangles. The idea is that if two triangles have the same three angles, then the ratio between the sides’ lengths is the same between triangles as well. For example, if two triangles have angles 45, 45 and 90 degrees, then the two sides other than the hypotenuse are equal in each triangle.
Measure the length of the shadow cast by the flagpole on a sunny day. Use a yardstick or meter stick to do this. Denote the length with the letter S, for “shadow.”
Plant a stick vertically in the ground near the flagpole. Measure the length of its shadow and denote it with the lowercase letter s, to stand for smaller shadow.
Measure the stick’s vertical height. Denote it with the letter h.
Calculate the flagpole’s height, H, using the formula H/S = h/s. In other words, H = h(S/s).
For example, if S is 15 feet, h is 4 feet (perhaps you used a yardstick as the stick) and s is 3 feet. Then H is 4*(15/3) = 20 feet tall. This is the height of the flagpole.
The straighter, taller and more vertical the stick, the better will be the accuracy of your h and s measurement. You can use a level to make sure the stick is vertical.