A tower has a vertical flagstaff fixed at its top. An observer standing somewhere on level ground sees the tower's top at an elevation of 30°, while the flagstaff's tip is seen at an elevation of 60°. The flagstaff's height and the tower's height are in the ratio:

Let the point be d m from the foot. The tower alone is d·tan30° = d/√3, and tower plus flagstaff is d·tan60° = d√3, so the flagstaff is d√3 − d/√3 = 2d/√3 — exactly twice the tower's height. The ratio 2 : 1 holds for every d, so no extra data is needed. Comparing the two total heights gives tan60°/tan30° = 3 : 1, which is the ratio of flagstaff-plus-tower to tower, not of flagstaff to tower.