A vertical flag mast stands on the top of a hill 50 m high. From a point on the plain in line with the foot of the hill, the angle of elevation of the top of the hill is 30° and that of the top of the mast is 45°. The height of the mast is:

The first sighting fixes the distance: d = 50·cot30° = 50√3 ≈ 86.60 m. The second sighting then puts the top of the mast at d·tan45° = 50√3 m above the plain, so the mast alone measures 50√3 − 50 = 50(√3 − 1) ≈ 36.60 m. The figure 50√3 ≈ 86.6 m is the whole height above the plain, hill included, rather than the mast; adding instead of subtracting the hill gives 50(√3 + 1) ≈ 136.6 m; and halving somewhere along the way gives 25(√3 − 1) ≈ 18.3 m.