Lecture notes
Area Computation
- Module
- Institution
- Book
Computing the area of parabola
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
Vertical Curves
Seller
Follow
ravenrvn
Content preview
GEFS-CE11 - (Fundamentals of Surveying)INSTR. MMMayos UNIVERSITY OF THE CORDILLERAS MODULE 3
METHODS OF DETERMINING AREA OF CLOSED TRAVERSE
A. AREA BY TRIANGLES
1. Using the definition of the area of a triangle
When the sides of the traverse have few sides and have a small area, this method
can be used. This method is based on the area of a triangle which is equal to one-
half of the product of its base and altitude.
P
θ
β 90 0
Q
N
M
Fig. A
1
𝐴𝑟𝑒𝑎 = (𝐵𝑎𝑠𝑒)(𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒)
2
1
𝐴 = 𝐵𝐻
2
But from right triangle MNP, solving for H
𝐻
𝑠𝑖𝑛 𝛽 =
𝑋
𝐻 = 𝑋𝑠𝑖𝑛 𝛽
Thus
1
𝐴 = 𝐵𝐻
2
𝐴 = 𝐵(𝑋𝑠𝑖𝑛 𝛽)
Or from right triangle QNP, solving for H
𝐻
sin 𝜃 =
𝑌
𝐻 = 𝑌𝑠𝑖𝑛 𝜃
1
𝐴 = 𝐵𝐻
2
1
𝐴= 𝐵(𝑌𝑠𝑖𝑛 𝜃)
2
, GEFS-CE11 - (Fundamentals of Surveying)
INSTR. MMMayos UNIVERSITY OF THE CORDILLERAS MODULE 3
Therefore, the area of a triangle can be determined as equal to one-half the
product of two adjacent sides multiplied by the sine of the included angle. In
Fig. A above, if the adjacent sides are X and B, the included angle is β, and if the
two adjacent sides are Y and B, the included angle is θ. This can be use when two
adjacent sides and the included angle are known.
2. Using Heron’s Formula
Heron’s formula can be use when the three sides of the triangle are known. In
the figure above (Fig. A), sides X, Y, and B are the sides, the area of the triangle
is,
𝐴= 𝑆(𝑆 − 𝑋)(𝑆 − 𝑌)(𝑆 − 𝐵)
Where:
1
𝑆= (𝐴 + 𝐵 + 𝑌)
2
The two methods mentioned above can be used to determine the area of a closed
traverse by dividing the closed traverse into a series of triangles.
Sample Problem no. 1
Given in the table below is the technical descriptions of a transit-tape survey of a
closed traverse. Determine the area of the closed traverse using triangles.
Course Distance Bearing Latitudes (m) Departures (m)
(m)
1–2 𝟐𝟗𝟖. 𝟒𝟒𝟔 S 36057’ E −238.506 +179.401
2–3 𝟒𝟎𝟑. 𝟒𝟕𝟒 S 43038’ W −292.022 – 278.413
3–4 𝟐𝟓𝟐. 𝟒𝟏𝟎 N 73016’W +72.674 – 241.721
4–5 𝟑𝟒𝟖. 𝟓𝟓𝟎 N 12010’ E +340.721 +73.459
5–1 𝟐𝟗𝟏. 𝟖𝟎𝟎 N 66020’E +117.133 +267.259
EL = 0.00 m ED = - 0.015 m
N
Final Figure 1
0
66 20’
5 36057’
2
0
12 10’ 0
43 38’
4 0
73 16’
3
Solution:
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller ravenrvn. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £5.79. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
73314 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now