Plane and Applied Surveying 2 Trigonometric Levelling theory -What is Trigonometric Levelling. -Measurement Using Trigonometry. Measurement Using Trigonometry. -The vertical angle and the slope distance between the two points are measured. -If You Are Able To Get to the base of the Tower Or The Building. Trigonometric Levelling I- If base of the object is accessible: 1. Instrument at station A is lower than station B. The three points (A, B, and O) are on the same vertical plane 2 Instrument at point B is lower than A. The three points (A, B, and O) are on the same vertical plane. 3. If the two instrument heights were at the same level. *Example: Find the vertical height of electrical column over a hill. The reading is taken from two instrument station (P, and R), and the horizontal distance between thereof is (60 m). The horizontal angle of RPQ = 60°30′ , and the horizontal angle of PRQ = 68°18′ . The vertical angle from P to Q =10°12′ , and the vertical angle from R to Q = 10°48′ . Find the reduced level of point Q if the reduced level of (B.M) = 435.065m and the staff reading from P and Rare (1.965, and 2.055) m respectively. And then check the result. Asst. Prof. Salar K.Hussein Mr. Kamal Y.Abdullah Asst.Lecturer. Dilveen H. Omar Erbil Polytechnic University Technical Engineering College Civil Engineering Department

1. 1
Erbil Polytechnic University
aErbil Polytechnic University
Technical Engineering College
Civil Engineering Department
l Engineering College
Civil Engineering Department
?
?
Prepared by
Assist. Prof. Salar Khudhur Hussein
Assist. Lecturer Mr. Kamal Yaseen
Plane and Applied Surveying 2

2. What is Trigonometric Levelling:
• Trigonometric leveling is the process of determining the differences
of elevation of stations from observed vertical angle and known
distance.
The vertical angle and the horizontal angle are measured by mean
of theodolite
Relative heights are calculated using trigonometric function
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3. Measurement Using Trigonometry
• Sometimes, it is hard to measure the height of an object because the object
is too tall to measure directly. However, by using math, it is possible to find
the height without measuring it directly. Trigonometry could be a way to find
the height without actually measuring the height
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4. Trigonometric Levelling:
Trigonometric leveling is used for the following cases:
1. Difficult terrain, such as mountainous areas, prevent the use of traditional differential
leveling.
2. The height difference is large but the horizontal distance is short such as height to tower
or a tall building.
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5. The vertical angle and the slope distance between the two points are
measured
• a. Electromagnetic distance measurers (EDM) is used to measure the slope
distance .
b. Theodolite is used to measure the vertical (or
zenith) angle.
• The purpose of trigonometric leveling is to find the invisible distance, where
cannot be measured directly, and it is also used to find the vertical distance.
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6. .
If You Are Able To Get to the base of the
Tower Or The Building
STEP 1
Set up your theodolite over a ground mark and take Vertical angle to the top of the
building and Vertical angle to the bottom of the building.
STEP 2
Measure the horizontal distance from the building to the ground mark (instrument
station).
STEP 3
Calculate the vertical distance(height). It is required to break the formula down in to
two right angle triangles.
As shown in the figure 1.
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7. Trigonometric Levelling
I- If base of the object is accessible:
• 𝑆1 = 𝐷 tan ∝
• 𝑆2 = 𝐷 tan 𝛽
• ℎ = 𝑆1 + 𝑆2
• ℎ = 𝐷 (tan ∝ + tan 𝛽) figure 1
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8. II- If base of object is not accessible:
In this case the instrument is fix at two different points.
It should be noted that the two instrument station (A,
and B) and the object(O) are at the same vertical line.
The following notes should be considered:
• Distance is unknown, cannot be measured.
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9. 𝑆 = 𝑟2 − 𝑟1 = 𝐻1 − 𝐻2
𝐻1 = 𝐷 tan ∝ ……………… (1)
𝐻2 = (𝑑 + 𝐷) tan𝛽 ……… (2)
Subtract equation 1 and 2
𝐻1 − 𝐻2 = 𝐷 tan ∝ − 𝑑 tan 𝛽 − 𝐷 tan 𝛽
𝑆 + 𝑑 tan 𝛽 = 𝐷 (tan ∝ − tan 𝛽)
𝐷 =
𝑆 + 𝑑 tan 𝛽
tan ∝ − tan 𝛽 9
II.1. Instrument at station A is lower than station B.
The three points (A, B, and O) are on the same vertical plane.
B A O

10. 𝑆 = 𝑟1 − 𝑟2 = 𝐻2 − 𝐻1
𝐻2 = (𝑑 + 𝐷) tan 𝛽 ……… (1)
𝐻1 = 𝐷 tan ∝ ……………… (2)
Subtract equation 1 and 2
𝐻2 − 𝐻1 = 𝐷 tan 𝛽 + 𝑑 tan 𝛽 − 𝐷 tan ∝
𝑆 − 𝑑 tan 𝛽 = 𝐷 (tan 𝛽 − tan ∝)
𝐷 =
𝑆 − 𝑑 tan 𝛽
tan 𝛽 − tan ∝ 10
II-2 Instrument at point B is lower than A.
The three points (A, B, and O) are on the same vertical plane.
B A O

11. II.3. If the two instrument heights were at the
same level.
𝐷 =
𝑑 tan 𝛽
tan ∝ − tan 𝛽
𝑅𝐿𝑜 = 𝐻. 𝐼 + 𝐻 Where S = 0
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12. If the base of object is not accessible:
The three instrument stations and object are not on the same vertical plane.
Distance (D1, and D2) between instrument stations and the object station are
unknown.
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13. Measure the horizontal distance between instrument station (A, and B)
Measure horizontal angle (θ1, and θ2)
θ1= OBA , θ2= BAO
Δ ABO’ θ = 180 – (θ1+ θ2)
Use (Sin Rule):
sin 𝜃
𝑑
=
sin 𝜃1
𝐷1
=
sin 𝜃2
𝐷2
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Plan

14. 14
Find D1, and D2
𝐻1 = 𝐷1 tan ∝ ……………… (1)
𝐻2 = 𝐷2 tan 𝛽 ……… (2)
𝑅𝐿𝑜 = 𝐻. 𝐼𝐴 + 𝐻1
To check: 𝐻. 𝐼𝐴 = 𝑅𝐿𝐵.𝑀 + 𝑟1

15. • Example:
Find the vertical height of electrical column over a hill. The reading is taken from two
instrument station (P, and R), and the horizontal distance between thereof is (60 m). The
horizontal angle of RPQ = 60°
30′
, and the horizontal angle of PRQ = 68°
18′
. The vertical
angle from P to Q =10°
12′
, and the vertical angle from R to Q = 10°
48′
.
Find the reduced level of point Q if the reduced level of (B.M) = 435.065m and the staff
reading from P and Rare (1.965, and 2.055) m respectively. And then check the result.
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16. Δ ABO’
θ = 180 – (θ1+ θ2)
θ = 180 – (68°
18′
+60°
30′
) = 51°
12′
Use (sin Rule):
sin 51°12′
𝑑
=
sin 68°18′
𝐷1
=
sin 60°30′
𝐷2
D1 = 71.5 m
D2 = 66.90 m
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17. 𝐻1 = 𝐷1 tan ∝ = 71.5 tan 10°
12′
= 12.86 m
𝑅𝐿𝑄 = 𝐻. 𝐼 + 𝐻1 = 𝑅𝐿𝐵.𝑀 + 𝑟1 + 𝐻1
= 435.065 + 1.965 + 12.86 = 449.89 𝑚
𝐻2 = 𝐷2 tan 𝛽 = 66.90 tan 10°
48′
= 12.76 𝑚
𝑅𝐿𝑜 = 𝐻. 𝐼𝐵 + 𝐻2 = 𝑅𝐿𝐵.𝑀 + 𝑟2 + 𝐻2
= 435.065 + 2.055 + 12.76 = 449.88 𝑚
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18. Thanks for all
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