UNIVERSITY OF NAIROBI
DEPARTMENT OF GEOSPATIAL AND SPACE TECHNOLOGY
By
Dr-Ing. S M Musyoka
USING A SCIENTIFIC CALCULATOR FOR SURVEY COMPUTATIONS
ANGULAR MEASUREMENTS
The following three units can be used as angular measures:
i.
Degrees
ii.
Radians
iii.
Grades
DEGREES
In the degree system (DMS), an angular measure is normally given as an integer part
called a degree, a fraction of a degree called a minute and a fraction of a minute called a
second (arc second). One (1) degree is equal to 60 minutes and one minute is equal to 60
arc seconds. The relationships are as follows:
Complete circle
= 360 degrees = 360*60 minutes = 360*60*60 seconds
1 degree
= 60 minutes = 3600 seconds
1 minute
= 60 seconds
We normally use symbols to represent the above measures. These are:
Degree ( ˚ )
Minute (́ ΄ )
Second ( ˝ )
Example;
The angles of a plane triangle are 60˚ 15΄ 12˝, 30˚ 45΄ 57˝, 88˚ 58΄ 51˝.The sum of these
quantities is done by first adding the arc second part and every 60 arc- seconds are
carried forward as 1 minute. Then add the minutes and every 60 minutes are carried
forward as 1 degree. The above sum is thus:
12˝ +57˝ +51˝ = 120˝ = 2΄ (carry 2΄ and write 0˝ )
15΄+45΄+58+2΄= 120΄ = 2˚ (carry 2˚ and write 0΄ )
60˚+30˚+88˚+2˚ = 180˚
The sum is therefore written as 180˚ 00΄ 00˝
In decimal degrees the above values:
15
12
60 

 60.25333333,
60 3600
45
57
30 

 30.76583333 ,
60 3600
1
58
51

 88.98083333 .
60 3600
Most scientific calculators have special keys to enable entry of the DMS values easily.
88 
The key normally has the symbols
˚΄˝
.
To enter the value 60˚ 15΄ 12˝,
˚΄˝
Type 60 then press
Then type 15 and press this button again,
Then type 12 and press this button again.
To convert the value entered into decimal degrees make use of the second function .
The DMS system is very popular in surveying and geodetic work. Most theodolites used
in the english speaking countries are graduated in this system.
RADIANS
A complete circle comprises 2Π radians. Thus 360˚ 00΄ 00˝ = 2Π. Generally, most
computer software use the radian as the angular measure.
GRADES
A complete circle comprises 400 grades. Thus 360˚ 00΄ 00˝ = 400 grades = 2Π. The
grades are not so much used here except in countries in Central and Eastern Europe.
Theodolites in these areas may be graduated in this system.
Scientific calculators have a key that enables conversion from one mode to the other.
BASIC COMPUTATIONS INVOLVING ANGLES
In positioning, we can express the coordinates of a point P either in the rectangular
system P(x,y) or in the polar system P (r, θ). See Figure 1 below.
In the rectangular system, if the coordinates of the origin are known, say Po(Xo, Yo); then
the coordinates of P(XP,YP) can be obtained by computing the components of values in
either axis. For example, the change in the x- axis (ΔX) or the change in the y- axis (ΔY).
Then the coordinates of P are
X P  X 0  X
YP  Y0  Y
2
x
Δy
Δx
θ
Po
P
r
y
Figure 1: Rectangular (x,y) and polar coordinates (r,θ).
The components ΔX and ΔY can be computed if the angle (bearing), θ and the distance r
have been measured. Thus
X  r cos
Y  r sin  .
Note that θ is always measured form the x-(north) axis in a clock wise manner. In
geodetic work the x- axis and y- axis are interchanged unlike the convention in
mathematics.
When θ is in other quadrants, apart from the first one, one has to be careful to get the
correct sign (positive or negative). Scientific calculators are equipped with an inbuilt
function that enables computation of ΔX and ΔY easily taking into consideration the
Rec(
Pol(
quadrant. The key representing this is normally marked
The REC( is the function you will need. For most CASIO calculators this function is
accessed by pressing the SHIFT function (normally in brown color) first.
Example 1: (Using CASIO fx-82MS calculator)
To get ΔX and ΔY given the polar coordinates r=234.456m, θ=62˚ 15΄ 28˝, press or type
SHIFT REC( 234.456 , 62˚ 15΄ 28˝ ) =
You get 109.1379506 as the ΔX and pressing RCL F you get the ΔY which is
207.5054787
The coordinates of P are then:
Xp = Xo + ΔX
Yp = Yo + ΔY,
Xo, Yo being the coordinates of Po.
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Example 1:
Given the coordinates of two points A and B, find the distance between them and the
bearing of B from A.
Station
A
B
Northing (m)
9 865 256.54
9 840 115.42
Easting (m)
256 345.87
255 112.11
The solution is found by pressing or typing the following:
POL((9840115.42-9865256.54),(255112.11-256345.87))=
The distance is displayed as 25 171.37419m and the bearing of B from A obtained by
pressing RCL followed by F buttons to get -177.1905556. Since a negative bearing is not
allowed, we have to add 360 to all negative cases. Thus our bearing is 182.8094444
degrees and pressing the degree button we get 182˚ 48΄ 34˝.
Note that the change in Northing part is entered first, followed by the change in Easting.
If the required bearing is from A to B then the subtraction is done as B-A for both
Northing and easting. The distance is displayed first in most CASIO calculators while the
bearing is saved in memory F and the distance in memory E.
Calculator Memory
Modern ordinary calculators have more than one memory. Most ordinary CASIO
calculators have memories marked A, B, C, D, E, F. Each memory storage carries one
value at a time and if a second value is saved, it automatically deletes the previous
memory contents and stores the latest value. Intermediate values of long calculations can
be stored in between computations for later use.
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Exercise 1:
Date assigned: 10/10/2012
Date due:
17/10/2012
1. Below is a table of UTM coordinates for some triangulation points in Kenya.
Station
Arle
89S13
Marop
Marigat
Ngelesha
Marmanet
Legisianan
SKP 103
Kwaibus
Northing (m)
60 985.01
73 209.64
56 284.91
54 326.86
49 916.36
19 762.29
2 242.91
19 754.16
32 584.14
Easting (m)
807 789.45
778 763.24
811 775.00
826 916.38
862 392.08
865 125.40
840 260.88
800010.19
834 263.90
Compute the bearing and distance of
a) Arle from 89S13
b) 89S13 from Arle
c) Marop from Marigat
d) Marigat from Marop
e) Ngelesha from Marmanet
f) Marmanet from Marop
g) Marmanet from Legisianan
h) Legisianan from Marmanet
2. Determine the coordinates of
a) Gobat whose bearing and distance from Legisianan are respectively 319˚ 50΄ 38˝ and
22 723.55m
b) SKP 101 whose bearing and distance from SKP 103 are respectively 321˚ 37΄ 19˝ and
21 133.24m
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