There may be two cases to find height of an object using a theodolite:
1. When he base of the object is accessible.
2. When the base of the object is inaccessible.
1. Base of the Object being Accessible (Fig. 9.13):
To find the height of the object above a Bench Mark (or above the instrument-station):
Let H = the height of the object above the B.M.
h = the height of the object above the instrument axis.
hs = height of the instrument axis above the B.M.
α = the vertical angle observed at the instrument-station.
D = the horizontal distance in metres measured from the instrument-station to the base of the object.
Then, h = D tan α
H = h + hs = D tan α + hs
If the height of the object above the instrument-station is to be found out, then add the height of the instrument axis to the height of the object above the instrument axis. The height of the instrument axis may be obtained in two ways.
(i) By measuring the height of centre of the eye-piece above the station point by a steel tape.
(ii) By readings the staff through the object-glass when held just near the eye-piece end.
2. Base of the Object being Inaccessible [Fig. 9.14 (a & b)]:
To find the height of the object above a bench mark (B.M.):
(i) Choose two stations A and B suitable on a fairly level ground so that they lie in a vertical plane passing through the object in line with the object, and measure the distance between them accurately.
(ii) Set up the instrument over the station. A and level it accurately.
(iii) With the altitude bubble central and with the vertical vernier reading zero, take a reading on the staff held on the B.M. or reference point.
(iv) Bisect the object P and read both verniers. Change the face, again sight P and read both verniers, Take mean of the four readings, which is the correct value of the vertical angle.
(v) Shift the instrument to B and take similar observations as at A.
Let α = the angle of elevation observed at A.
β = the angle of elevation observed at B.
b = the horizontal distance between the instrument-stations A and B.
D = the distance of the object from the near station.
h = height of the object P above instrument axis at A’.
ha =the staff reading at the B.M. when the instrument is at A.
hb = the staff reading at the B.M. when the instrument is at B.
hd = the level difference between the two positions of the instrument axis.
= ha – hb
(a) When the Instrument at farther station B is higher than that the near station A (Fig. 9.14 a):
(b) When the instrument at father station B is lower than that at the near station A (Fig 9.14b):