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Neenu
February 17, 2021

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)

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.

- 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''.
- 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
**AO**_{1}B. - Next sight the ranging rod at C , then O2 and close the traverse. The angle obtained is
**AO**and_{1}C**AO**_{1}O_{2} - Bring the telescope to A, by rotating in the same direction.
- Repeat the steps by changing the face of the theodolite to left Face.
- The average of horizontal angles is measured from the obtained readings.
Area of a Triangle Using Theodolite-Horizontal Angle Figure-2

You can write down the angle **BOC **= AO_{1}C - AO_{1}B and CO_{1}O2 = A1O_{1}O_{2}- AO_{1}C. Let, angle AO_{1}B = ፀ1 ; BO_{1}C = ፀ2; CO_{1}O2 = ፀ3.

- Now after the measurement of horizontal angles from O
_{1}, start measuring the vertical angles to A, B and C. Here, the telescope will be in face left position. - 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.
- Repeat the procedure from face right.
- 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.

Consider the Triangle O_{1}AO_{2}

The angle O_{1}AO_{2}= 180 - (θ_{1}+
θ_{2}+ θ_{3}+ θ_{4})

Now, Apply Sine Rule to the Triangle O1AO2,

**O _{1}O_{2}/ sin ( angleO_{1}A) = O_{1}A/sin(θ_{4})**

From this, the unknown O_{1}**A **is determined.

Similarly, consider triangles O_{1}BO_{2}and O_{1}CO_{2} to determine O_{1}B and O_{1}C using sine rule.

__Step 2: Determination of distance AB, BC and AC__

Consider triangle O_{1}AB

Apply cosine rule on the triangle:

Hence,

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)]

If, the vertical angle obtained by sighting point A from station O1 is α_{1 }and h1 is the vertical distance of point A measured from the line of sight of theodolite, then

where, D = O_{1}A;

Similarly, the vertical distance of point B and C, is obtained as h2 =**D tan α2** , where D= O_{1}B ;

h3 = **D tan α3; **where D= O_{1}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;

Read Also:

- 1. Surveying-Definition, Principle and Classification
- 2. Principles of Surveying
- 3. Linear measurement in surveying
- 4. Reconnaissance Survey and Index Sketching
- 5. Types of Survey based on Instruments
- 6. What is Levelling in Surveying?
- 7. What is chain surveying?
- 8. Equipment Used in Chain Surveying
- 9. Type of Chains Used in Chain Survey

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