Mathematics · Lesson 22

Longitude & latitude (earth as a sphere)

JAMBWAECNECO

What the syllabus asks

JAMB: Geometry/mensuration: longitude and latitude, distances along lines of latitude and longitude

WAEC: Mensuration: the earth as a sphere: great and small circles, latitude and longitude, distance along great circles (meridians, equator) and small circles (parallels of latitude) in km and nautical miles, radius of a parallel, longitude and time

NECO: Mensuration: the earth as a sphere: great and small circles, latitude and longitude, distance along great circles (meridians, equator) and small circles (parallels of latitude) in km and nautical miles, radius of a parallel, longitude and time

What you will learn

Describe the positions of places using latitude and longitude.
Find differences in latitude and longitude.
Calculate distances along meridians and the equator.
Calculate the radius of a parallel and distances along it.
Express distances in nautical miles and use knots.
Solve problems on longitude and time, and find new positions after travel.

Key points

Latitude: N/S of equator; longitude: E/W of Greenwich.
Same side: subtract angles; opposite sides: add.
Great circles (meridians, equator) have radius R; parallels have radius R cos α.
Meridian or equator distance: θ/360 × 2πR.
Parallel distance: θ/360 × 2πR cos α, with θ the longitude difference.
Nautical miles: 60θ on great circles; 60θ cos α on parallels; 1 knot = 1 n.m. per hour.
Time: 15° per hour, 4 minutes per degree; East is ahead.
Use the R and π given in the question; draw side and top views.

Start here: The earth as a sphere

For calculation, we treat the earth as a sphere of radius about 6400 km (WAEC usually states R = 6400 km and π = 22/7; use exactly what the question gives). Its axis passes through the North and South Poles, and it rotates about this axis once a day.

Two kinds of circles can be drawn on a sphere:

A great circle has its centre at the centre of the earth. Its radius is R. The equator and every meridian (with its opposite meridian) form great circles.
A small circle has its centre elsewhere on the axis. The parallels of latitude (other than the equator) are small circles, and they get smaller towards the poles.

The shortest distance between two points on a sphere is along the great circle through them.

Worked example 22.1

Find the circumference of the earth along the equator, taking R = 6400 km and π = 22/7.

C = 2πR = 2 × 22/7 × 6400.
C ≈ 40,229 km (more precisely 40,228.6 km).

≈ 40,229 km

With the true value of π, 2π × 6400 ≈ 40,212 km. Small differences like this are why you must use the π stated in the question.

⚠️ Common mistake: thinking every circle on the earth has radius 6400 km. Only great circles do.

Try it yourself. Is the parallel 30°N a great circle?

Show solution

No; it is a small circle. Only the equator is a great circle among the parallels.

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Practice questions

1. Find the difference in longitude between 35°E and 25°W.

Show worked answer

60°.

2. Find the distance along the meridian between (15°N, 6°E) and (25°S, 6°E). (R = 6400 km, π = 22/7)

Show worked answer

θ = 40°: ≈ 4469.8 km.

3. Find the radius of the parallel 60°S.

Show worked answer

6400 × 0.5 = 3200 km.

4. Find the distance along the parallel 60°N between 20°E and 70°E (R = 6400 km, π = 22/7).

Show worked answer

50/360 × 40228.6 × 0.5 = ≈ 2793.7 km.

5. Find, in n.m., the distance along the equator between 10°W and 40°E.

Show worked answer

60 × 50 = 3000 n.m.

Key words

Great circle: A circle on the sphere whose centre is the centre of the earth; radius R.
Small circle: A circle on the sphere whose centre is not the earth’s centre.
Equator: The great circle at latitude 0°.
Meridian: A half great circle from the North Pole to the South Pole; all points on it share the same longitude.
Greenwich meridian: The meridian of longitude 0° (prime meridian).
Parallel of latitude: A small circle joining points with the same latitude.
Latitude: Angle north or south of the equator, measured at the earth’s centre.
Longitude: Angle east or west of the Greenwich meridian.
Nautical mile: Length of an arc subtending 1 minute (1/60°) at the centre of the earth along a great circle.
Knot: A speed of 1 nautical mile per hour.
Radius of a parallel: r = R cos α, where α is the latitude.

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