You can determine the height of a building without having to leave the ground, just by using simple trigonometric or geometric analysis. You can either use the building’s shadow, when the sun is high on a sunny day, or you can use a sextant to measure the angle to the top of the building. The former approach may be far more accurate, unless you have access to a very precise, mounted surveyor’s sextant.
Wait for a day when the sun is high enough so that the top of the building casts a shadow all the way down to the ground (as opposed to hitting the building on the other side of the street).
Place a straight stick (such as a meter stick) vertically in the ground. If "P" is the point on the ground where the shadow of the top of the building lands, you should position the stick a little closer to the building than that point P. The vertical stick should be mostly in the shadow of the building, with the top of the building casting a shadow some distance up the stick.
Measure the distance up the stick where the shadow of the top of the building stops (call this distance "A"). Measure the distance between the bottom of the vertical stick and point P, where the building’s shadow ends on the ground (call this distance "B"). Measure B in the same units as A. Measure the distance from point P to the base of the building (call this distance "C"). A laser meter may help you measure this distance, since the building might be quite far from point P. Note that the triangle made by P, A and B is similar to the triangle made by C, P and the top of the building. By the rule of similar triangles, the ratio of A to B equals the ratio of the height of the building to C.
Put measures A and B in the same units, so their units cancel out upon division. Divide A by B and multiply by C. This is the height of the building, in the units in which you measured distance C.
If the top of the building is significantly tapered, then the measure of C will be underestimated, and so will the height of the building. You should add to your measurement of C the extra distance inside of the building to reach the point directly under the top of the building that was casting the shadow at point P. That way, the small triangle made by A, B and P will be similar to the large triangle made by P, C and the height of the building.