Spherical Geometry

SUBJECT: MATHEMATICS

 

CLASS: SS 3

 

TERM: FIRST TERM

 

 

 

 

 

WEEK 7                                             Date:……………..

Topic

Spherical Geometry

–The Earth Description

– Concept of Longitude and Latitude

– Concept of Great and Small Circles

– Radii and length of Latitude

Description of Earth: The earth is approximately spherical in shape. It has the North pole to the extreme on the upper part and the south pole to the extreme on the lower part .  The poles are flat at the two extreme so that the shape is not circular.

The earth is divided into two halves by the equator. The equator defines the plane of revolution of the earth which means that it lies in a plane perpendicular to old axis and through its centre

The diameter of the earth is about 12,444 kilometres or around 7,918 miles. It can be approximated as an ellipsoid

The concept of longitude and latitude is used to describe the location of an object on earth’s surface. Longitude measures east-west distance, while latitude measures north-south distance. The longitude is measured in degrees from 0 to 360, with 0 representing the Prime Meridian (0 degrees). Similarly,

The imaginary line (straight) through the centre running from the North pole to the South pole is called the axis of the earth. The equator is the largest imaginary circle that runs around the centre of the earth.

On the other hand, the radius of the sphere is not constant. The radius reduces from the equator towards the poles. The radius of the earth therefore changes from 6360km to 6380km and it is approximated to 6400km . Orange is another example of a sphere.

 

LATITUDE AND LONGITUDES

These are imaginary lines that run through the surfaces of the earth to locate position of a place on the earth’s surface.  The horizontal plane through the centre of the earth and perpendicular to the axis of the earth forms a boundary line which is a circle. This circle is called a Great circle And this particular circle is the Equator. All great circles have radii equal to the radius of the earth, other examples of great circles are all lines of longitudes that run from the North pole to the South Pole, This great circle is called a meridian. The equator is a reference latitude while the inference longitude is the prime meridian.

Small circles are the lines of latitude that run from East to West except the equator. The lines of latitude are also referred to as parallel of latitude because they are parallel to each other. Half a meridian is a longitude.

 

The prime meridian is designated as longitude 0^o since it is a reference longitude. Longitudes to the left of the prime meridian are said to be west of the prime meridian, while longitudes to the right are said to be East of the Prime Meridian.

 

Prime Meridian passes through the city of Tema in Ghana, and a place in London called Greenwich, and for this reason, it is sometimes called the Greenwich Meridian .

 

NGSK represents the prime Meridian. NBS, NFS, NHS and NQs are half a meridian.  NQs, NFS, NHS and NBS represent the parallels that run from east to West.

The concept of longitude and latitude is used to describe the location of an object on earth’s surface. Longitude measures east-west distance, while latitude measures north-south distance. The longitude is measured in degrees from 0 to 360, with 0 representing the Prime Meridian (0 degrees). Similarly,

The diameter of the earth is about 12,444 kilometres or around 7,918 miles. It can be approximated as an ellipsoid

The spherical geometry of Earth plays a major role in navigation and global communication, and has been studied by scientists for centuries. The diameter of the Earth is about 12,444 kilometers, which makes it a fairly large sphere compared to other planets in our solar system. While we can’t say for certain what Earth’s exact radius is, scientists have calculated that it is approximately 6400 kilometers at the equator and slightly smaller near the poles.

 

LINES OF LATITUDE

 

Cross section view.

In fig 3(a) HBG is the equator, EAF is a parallel of latitude north of the Equator while ICJ is a parallel to latitude south of the Equator.

The minor arc AB from fig 3(b) subtends angle ADB =   at the centre of the earth; and because of this, the point A is said to have a latitude  x^o north of the Equator. Similarly, the minor arc BC subtends angle COB = B  at the centre of the Earth and because of this, the point O is said to have a

AOB = x  AOC = B.

 

LINES OF LONGITUDES

In figure 4 (a) below NQS is the Prime Meridian, NPS is the longitude West of the Prime Meridian, while NRS is the longitude East of the Prime Meridian.

 

The Prime Meridian cuts the Equator at Q, the meridian NPS cuts the Equator at P while the Meridian NRS cuts the equator at R. O is the centre of the Earth.

From figure 4b , the minor arc PQ subtends angle POQ = θ^o at the centre of the circle. P is said to lie on the longitude θ^o, West of the prime meridian. Similarly, the minor arc RQ subtends angle ROQ = θ^o₂ at the centre of the circle.

So, < POQ = θ       <QOR = θ

The latitude and longitude of a place on the surface of the Earth are written as an ordered pair. For instance the city of Lagos in Nigeria is on (Latitude 6^oN, longitude 3^o E). this can be written as (6^oN, 3^oE).

 

Example 1

The city of Kumasi in Ghana is on latitude 6oN , longitude 10W ). Express in a shorter form.

Solution.

Kumasi on (latitude 6^oN,longitude 1^oW) = K ( 6^oN, 1^oW)

 

Example 2.

Draw sketches to illustrate the positions of

i.             Kinshasha on (16^oS, 25^oE )

ii.            Libya on (35^oN, 42^oW)

Solution

i. Kinshasha ( 16^oS, 25^oE)

 

 

 

 

 

 

 

 

 

Evaluation

1. G is the point where the Greenwich Meridian crosses the equator. Lines of latitude 70^oN and 30^oS and longitude 80^oE and 40^o W are given, make a sketch of the positions on the earth surface.

2. State the latitude and longitude of the following points.

(a) P                (b) Q                           (c ) R               (d) X    (e ) Y             (f ) Z

 

 

Reading Assignment:

New General Mathematics, chapter 7 pgs 52- 55.

Essential Mathematics for SS3, by AJS Oluwasanmi chapter 7, pgs 94-96

 

General Evaluation

X  and Y are points on points on the Earth s surface at opposite ends of a diameter through the centre of the centre of the Earth.(a)If the longitude of  X is 19⁰E,what is the longitude of Y.

(b)If the latitude of X is 52⁰N,what is the latitude of Y?

 

Weekend Assignment

1. The parallel of latitude is also referred to as …………………

(a) horizontal of latitude, (b) verticals of latitude   (c) lines of latitude  (d) meridians.

2. Which of the following is odd in the option below:

(a) Equator                         (b) Prime meridian        (c) latitude  0o (d) parallel of latitude

3. The radius of an equator is equal to the radius of a prime meridian .

(a) True                 (b) False                     (c) Neither false nor true  (d) incomplete information.

 

 

 

 

 

 

 

 

Use this diagram for question 4 and 5.

4.  P is located on ………………………

(a)  (45^oN, 20^oE)   (b) (45^oS, 20^oW ) ( c)  (45^oN. 20^oE )    (d) 45^oS, 20^oE).

5.   Q is located on ……………………

(a) (20^oN, 200E )           (b)  (30^oN, 20^oE)               (c) (30^oS, 20^oE )  (d) 30^oS, 20^oW)

6. State the latitude and longitude of A

(70oN, 20oW) (b) (20oS, 40oE)———-

(c) 40oS, 30oW (d) 70oS , 30oE—————

7. State the Latitude and longitudes of B

(a) 80oN , 30oE ———————————————–(b) 30oS, 50oW———-

8. State the latitude and longitudes of CÂ

  Â(c) 70oS, 20oE ——- ================================================ (d) 20oN, 40oW——-Â

9. The latitude of the point marked X is ___________ and its longitude is __________

Â

(a) 30oS, 60oW (b) 20oS, 80oW (c) 30oS, 60oE (d) 20oS , 80oE

10.The latitude of the point marked Y is __________ and its longitude is ___________

(a) 60oN, 70oE (b) 40oS, 50oW (c) 30oS , 80 oE (d ) 20oS, 90oWÂ

 

THEORY

1. Make a rough sketch of a globe on your sketch, mark the following campuses of Good Shepherd Schools

i. Diligence Campus ( 60⁰N, 55⁰W) and . Wisdom campus ( 35^oS, 55^oW).

ii. Peace campus (60⁰N, 20⁰E ) and Alakuko campus ( 35⁰S, 20⁰E).

1.           In the 18^th century the length of ¼ of the earth’s circumference was taken to be 1000km. Use this value to calculate the radius of the earth to 3.s.f.

Solution

Radius of the earth = 3,600km ≈ 3.6 × 1,000 km = 3,600,000 km

2. Latitude is measured in degrees from the equator (0o). Express the latitude of 020N and 210S in radians. How does it compare to the value of the circumference of a circle?

Latitude 020N = 20/360 x 2π radians ≈ 0.834 radians

Latitude 210S = 210/360 x 2π radians ≈ 1.154 radians

The values for latitude are slightly less than half that of the circumference of a circle, which is 2π radians. This reflects the fact that latitude is defined as the angle between a line running through the center of the earth and the equator, while circumference is based on an arc formed by a great circle passing through both poles. As such, latitude represents only part of the total circumference.