Determination of Area and Level Difference Between Points in a Inclined Plane Using Theodolite Surveying

To solve this problem, we will find the area between three inaccessible points A, B and C that lies over an incline plane. As these points lie over an inclined plane, each point is at a different height level compared to the other point. Hence, we need to determine the level difference between the points. (Figure-1)


To proceed with the work, we can make use of a Theodolite with tripod for measuring the horizontal and vertical angle, Ranging rod to locate each points, Pegs and Tapes  to measure the linear distance between the points.


The basic principle followed is to determine the linear distance between the points by measuring the horizontal angles and the level difference by measuring the vertical angles. For this:

1. Two stations O1 and O2 are fixed at a known distance from points A,B and C. The distance O1O2 is known.

2. The horizontal angles to point A,B,C from the station O1 and O2, can give the area of the triangle.

3. The vertical angles to point A,B, C from the stations O1 and O2, gives the level difference between the points. If the height of A is h1 with respect to the line of sight, and the vertical angle is ፀ4, then

h1 = D tan ፀ4 (Figure-2)

Procedure to Determine Horizontal and Vertical Angles to the Points in a Inclined Plane

Start working by drawing a rough figure on your working pad. We need to determine the horizontal and vertical angles from stations O1 and O2 to the points A,B and C. Here, keep in mind that the points O1 and O2 are know to us, that O1O2 distance is measurable and can be used for further calculations.
  1. To start calculating the horizontal angles, set the theodolite with tripod on the station O1, and keeping the ranging rod at A. Perform all the temporary adjustments for face right. Sight to A with the help of a lower clamp screw. Bisect accurately with the help of a lower tangent screw. Note the readings on the Vernier's A and B as 0 00'00''.
  2. Now, sight to B with the help of an upper clamp screw and bisect precisely using upper tangent screw. Note the readings on verniers A and B which will give the angle AO1B.
  3. Next sight the ranging rod at C ,  then O2 and close the traverse. The angle obtained is AO1C and AO1O2
  4. Bring the telescope to A, by rotating in the same direction.
  5. Repeat the steps by changing the face of the theodolite to left Face.
  6. The average of horizontal angles is measured from the obtained readings.
    Area of a Triangle Using Theodolite-Horizontal Angle
You can write down the angle BOC = AO1C - AO1B and CO1O2 = A1O1O2- AO1C. Let, angle AO1B = ፀ1 ; BO1C = ፀ2; CO1O2 = ፀ3.
  1. Now after the measurement of horizontal angles from O1,  start measuring the vertical angles to A, B and C. Here, the telescope will be in face left position.
  2. With the help of lower clamp and vertical clamp, sight to A, B and C. Use the lower and vertical tangent screws to bisect accurately. Here, the verniers C and D are used to measure the vertical angles to A,B and C from the station O.
  3. Repeat the procedure from face right.
  4. Determine the average vertical angles from the face left and face right position.
    Determination of Height of a point using theodolite
Once you measured the horizontal angles and vertical angles from station O1, shift the theodolite to station O2 and determine the horizontal angles to A, B and C in both face left and face right position. You need to determine the vertical angles from station 2, as we have already got the solution from station O1.


1. Determination of Straight Distance between Points- A'B', B'C' and A'C'.

Step 1: Determination of distance of points A,B and C from O1, i.e. O1A, O1B and O1C.

Consider the Triangle O1AO2

The angle O1AO2= 180 - (θ1+ θ2+ θ3+ θ4

Now, Apply Sine Rule to the Triangle O1AO2, 

O1O2/ sin ( angleO1A) = O1A/sin(θ4)

From this, the unknown O1A is determined. 

Similarly, consider triangles O1BO2and O1CO2 to determine O1B and O1C using sine rule.

Step 2: Determination of distance AB, BC and AC

Consider triangle O1AB

Apply cosine rule on the triangle:


Similarly, consider triangles O1BC and O1AC and determine the sides BC and AC.

2. Determination of Inclined Distance AB, BC and CA.

AB = Sqrt of [ square of (h1-h2) + square of (A'B)] '
BC = Sqrt of [ square of (h2-h3) + square of (B'C)]
AC= Sqrt of [ square of (h1-h3) + square of (A'C)]

3. Determination of Level Difference Between the Points

Step 1: Determination of vertical distance of Points A, B and C

If, the vertical angle obtained by sighting point A from station O1 is αand h1 is the vertical distance of point A measured from the line of sight of theodolite, then 
 h1 = D tan α1

where, D = O1A;

Similarly, the vertical distance of point B and C, is obtained as h2 =D tan α2  , where D= O1B ;
 h3 = D tan α3; where D= O1C

Step 2: Level Difference Between the Points A, B and C

Level Difference of A and B =  h1 ± h2;
Level Difference of B and C = h2 ± h3;
Level Difference of C and A = h1 ± h3;

4. Determination of Area of triangle formed by Sides AB, BC and AC

From the above formula, the area of triangle can be determined. Where AB, BC and CA are a,b and c respectively.

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