Measuring the Heights of buildings using Trigonometric Leveling.

Object: Measuring the height of a building by trigonometric leveling.

Introduction.

• Trigonometric leveling is the process of determining the differences in elevations of stations using observed vertical angles and known distances, which are assumed to be either horizontal or geodetic lengths at mean sea level.
• The vertical angles may be measured using an accurate theodolite and the horizontal distances using a tachometer.

CASES:

There are two cases of Trigonometric leveling in our discussion.

1. Determination of height of an elevated object when the base of the object is accessible (reachable)
2. Determination of the object’s height when its base and top are visible but inaccessible (reachable).

CASE 1: DETERMINATION OF HEIGHT OF ELEVATED OBJECT WHEN BASE OF THE OBJECT IS ACCESSIBLE:

Let:

A = instrument station

P = point to be observed

C = center of the object

P’ = projection of P on a horizontal plane through C

D = horizontal distance CP’ between A and P

h’ = height of the instrument at A

h = height in between P and P’ i.e. PP’

θ = angle of elevation from A to P

1. 
   From triangle CPP’,
   

h = D tan θ

PROCEDURE:

1. Set up the theodolite at A and level it accurately concerning the bubble.
2. Direct the telescope towards P and bisect it accurately.
3. With the horizontal angle 0˚, read the vertical angle θ.
4. Find the stadia readings to measure the distance D, which should be the same as that of the distance measured by the help of tape in between A to p’.
5. Put the values in the formula and calculate the height.
6. This method is usually employed when the distance is small.

CASE 2: DETERMINATION OF HEIGHT OF ELEVATED OBJECT WHEN ITS BASE AND TOP ARE VISIBLE BUT NOT ACCESSIBLE:

Let:

P = point to be observed

P’ = projection of P on a horizontal plane.

A = first instrument station

θ = angle of elevation from A to P

B = second instrument station

α = horizontal angle from A to B

β = angle of elevation from B to P’

h = height of the instrument at A

h’ = height of instrument from P to P’ ( PP’)

b = distance between A and P’

D = distance from A to B

PROCEDURE:

1. Set the instrument at station A and level it accurately.
2. Direct the telescope towards P and bisect it.
3. By keeping the horizontal angle at 0˚, measure the vertical angle θ
4. Choose another station B in such a way that ABP forms a triangle.
5. Rotate the telescope towards B and note the horizontal angle α.
6. Put the staff rod at B and find the stadia readings.
7. Measure the distance D from A to B by tape which should be the same as that measured by the instrument.
8. Now, set the instrument at B and level it properly.
9. Direct the instrument towards P and bisect it.
10. Set the horizontal angle to 0˚ and measure the vertical angle β.
11. Find out the height by using the formula:

h = b tan θ

Where b = D sin β/sin (180˚ – α – β)

I.e. By sin rule

b/sin β = D/sin(180˚ – α – β)