**TOPIC 7: THE EARTH AS THE SPHERE ~ MATHEMATICS FORM 3**

**Features and Location of Places**

The Equator, Great Circle, Small Circles, Meridian, Latitudes and Longitudes

Describe the equator, great circle, small circles, meridian, latitudes and longitudes

**Definition of latitude and longitude**

The earth rotates about its axis, which stretches from the north to the South Pole.

**Latitude**and Circles round the Earth which go through the poles are circles of

**or**

*Longitude*

*meridians*.**is defined relative to the equator, which is the circle of latitude round the middle of the Earth while**

*Latitude***is defined relative to the circle of longitude which passes through Greenwich in London (**

*Longitude***Greenwich meridian).**

latitude of a position tells us how far north or south of the equator

it is while the longitude of a position shows us how far east or west of

the Greenwich meridian it is.

**L**

**atitude;**If we draw a line from the centre of the Earth to any position

*P*, then the angle between this line and the plane of the equator is the latitude of P.

**Longitude:**This is the angle between the plane through the circle of any Longitude P and the plane of the Greenwich meridian .

**can be either North or South of the equator while**

*Latitude***Longitude**can be either East or West of Greenwich.

es Salaam has latitude 7°S (i.e. 7° south of the equator) and longitude

39°E (i.e. 39° east of the Greenwich meridian). So Dar es Salaam is at

(7°S, 39°E).

**;**

*NB*Greenwich itself has latitude 51°N ( i.e. 51° north of the equator)and

longitude 0° (by definition). Johannesburg has latitude 26°S (i.e. 26

south of the equator) and longitude 28°E(i.e. 28°east of the Greenwich

meridian), therefore Johannesburg is at (26°S, 28°E).The north pole has

latitude 90°S but its longitude is not defined. ( Every circle of

longitude goes through the north pole).The south pole has latitude 90°s.

Its longitude is not defined.

*So all points on the equator (such as Nanyuki in Kenya) have latitude 0*

*°***Ranges; Latitude**varies between 90°S (at the south pole) to 90°N (at the north pole).

**Ranges; Latitude**varies between 90°S (at the south pole) to 90°N (at the north pole).

**Longitude**varies between 180°E and 180°W. These are the longitudes on the opposite side of the Earth from Greenwich.

*GREAT AND SMALL CIRCLES:*There is an essential difference between latitude and longitude.

Circles of longitude all have equal circumference. Circles of latitude

get smaller as they approach the poles. The centre of a circle of

longitude is at the centre of the earth. They are called

*great circles**.*For circles of latitude, only the equator itself is a great circle. Circles of latitude are called

*small circles.*

*Solution;*- (10°N, 30°E)
- (20°N, 20°W)
- (0°, 20°W)

- (40°S, 30°E)
- (50°S,20°W)
- (10°N,40°W)
- (40°N, 30°E)
- (80°N,10°E)
- (0°,0°)

*Difference between angles of latitude or longitude*If both points are south of the equator subtract the latitudes

If both points are the north of the equator subtract the latitudes

If one point is south of the equator and the other north then *add *the latitudes

If both points are east of Greenwich subtract the Longitudes

If both points are west of Greenwich subtract the Longitudes

If one point is east of Greenwich and the other west then *add *the longitudes

*A*and

*B*are on the same longitude, then the difference in latitude is the

**angle**subtended by

*AB*at the centre of the earth.

*B*are on the same latitude.

**angle**subtended by

*AB*on the earth’s axis.

places on longitude 30°E are Alexandria (in Egypt) at (31°N, 30°E),

Kigali (in Rwanda) at (2°S, 30°) and Pietermaritzburg (in South Africa)

at (30°S, 30°E).

- Kigali and Pietermaritzburg
- Kigali and Alexandria

*Solution*^{0}

*The difference is*33

^{0}

plane starts at Chileka airport (in Malawi) which is at (16°S, 35°E).

It flies west for 50°. What is its new latitude and longitude?

*Solution*- The difference in longitude between A and B
- The difference in longitude between D and E

**Distances along Small Circles**

P and Q are places west or east of each other, i.e they lie on the same

circle of latitude. Then when you travel due east or west from P to Q

you travel along an arc of the circle of latitude.

situation here is slightly different from that of the previous section.

While circles of longitude all have the same length, circles of

latitude get smaller as they get nearer the poles.

*Solution:*

*3,950 nm.*

*Navigation**Suppose*

a ship is sailing in a sea current, or that a plane is flying in a

wind. Then the course set the ship or plane is not the direction that it

will move in. the actual direction and speed can be found either by

scale or by the use of Pythagoras’s theorem and trigonometry.

a ship is sailing in a sea current, or that a plane is flying in a

wind. Then the course set the ship or plane is not the direction that it

will move in. the actual direction and speed can be found either by

scale or by the use of Pythagoras’s theorem and trigonometry.

*Draw*

the line representing the motion of the ship relative to the water. At

the end of this line draw a line representing the current. Draw the

third side of the triangle. This side, shown with a double – headed

arrow, is the actual course of ship.

the line representing the motion of the ship relative to the water. At

the end of this line draw a line representing the current. Draw the

third side of the triangle. This side, shown with a double – headed

arrow, is the actual course of ship.

ship sets course due east. In still water the ship can sail at 15km/hr.

There is a current following due south of 4kkm/hr. use a scale drawing

to find.

- The speed of the ship
- The bearing of the sip.

*Solution:*one hour the ship sails 15km east relative to the water. Draw a

horizontal line of length 15cm. In one hour the current pulls the ship

4km south. At the end of the horizontal line, draw a vertical line of

length 4cm.

- What course should be set?
- How long will the ship take to cover 120km?

*Solution*small island is 200km away on a bearing of 075°. A ship sails on a

bearing of 070°.Find the closest that the ship is to the island.

Exercise 5

5. Find thedistance in nau

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