Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 41

Table 2.3 Determining the Mass and Volume of the Earth

Determine the length of the polar circumference a.

Knowing that the circumference of a circle is b.

2* π *R
where R is the radius of the Earth

[R ] km = [a] km
(2 *

km π
[ R m ] meters = [ R km ]* 1000
Mass

)

c.

Knowing that mass is

(acceleration * R )

2

Mass =

G
where acceleration due to gravity is 9.8 m/s 2 , and G , the constant of proportionality,
is 6.67 × 10 - 11 m 3 /kg s 2
You can determine the mass (kg) of the Earth using the Earth's radius [ R m ] from [b].
To calculate the mass % error (assuming a true mass of 5.98 × 10 24 kg) d.

|[c] kg − 5.98 × 10

24

kg|

% error

= * 100

5.98 × 10

24

kg

Volume e.

Knowing that volume is
Volume = 4/3 * π * R 3
You can determine the volume (km 3 ) of the Earth using the Earth's radius [ R km ] from [b].
To calculate the volume % error (assuming a true volume of 1.0975095 × 10 12 km 3 ) f.

e

| km

3

− 1.0975095 × 10 km |

12 3

%error = * 100

1.0975095 × 10 km

12 3

60 minutes in a degree and 60 seconds in a minute. Consider the value 42.53 as
equal to 42 + 5/10 + 3/100. Convert the denominators of the fractions to 60 and
to 3600. Do so by multiplying each proper fraction by the number 1, written in
one case as 6/6 and in the other case as 36/36. In the case of the minutes, 5/10
is the same as 30/60. In the case of the seconds, 3/100 is the same as 108/3600.
Thus, 42.53 = 42 + 30/60 + 108/3600 and this value translates to 42 degrees, 30
minutes, 108 seconds. But, 108 seconds reduces to one minute and 48 seconds.
So, the answer is 42 degrees, 31 minutes, 48 seconds. The discrepancy in value
in reversing the process is due to round off. The method is accurate.

In the last paragraph, we multiplied fractions by 1, written one time as
6/6 and another time as 36/36. Creative use of multiplying by the number
1 is the root of addition of fractions. Often, unfortunately, one learns to
add fractions before learning to multiply them (perhaps as a simple cur-
ricular extension of how one learns to work with integer arithmetic). That