Longitude and Latitude II – SS3 Mathematics Lesson Note (Third Term)

The great circles on the earth are the lines of longitude and the equator. The distance between two places along a great circle is found by locating the places, working out their angular difference or sum, and applying the distance formula with the earth taken as a sphere of radius 6400 km.

Behavioural objectives

By the end of the lesson, students should be able to:

1. locate places on the earth’s surface;
2. calculate the angular sum or difference of those places;
3. apply the right formula;
4. calculate the distance between the points along the great circle.

Instructional resource

Model of the earth (globe).

What are great circles and small circles?

Mode: whole class.

Teacher’s activities: The teacher asks the students to look at the globe and say which lines on the earth’s surface are great circles and which are small circles.

Students’ activities: The great circles are the lines of longitude and the equator. The small circles are the parallels of latitude.

The teacher then explains that distances along great circles are distances along the lines of longitude and along the equator.

How do you find the angular difference or sum between two places?

Mode: whole class.

Teacher’s activities: The teacher leads the students to locate points and find their angular difference or sum.

Example 1

An aeroplane flies from a town P(40°N, 38°E) to another town Q(40°N, 22°W). It later flies to a third town T(28°N, 22°W). Calculate the angular difference or sum between:

1. P and Q
2. Q and T

Solution

1. P and Q lie on opposite sides of the Greenwich meridian (E and W), so the angular sum is 38° + 22° = 60°.
2. Q and T lie on the same side of the equator (both N), so the angular difference is 40° – 28° = 12°.

Example 2

Two places A and B lie on the same parallel of latitude 34.6°N. Their longitudes are 38.7°E and 22.3°W respectively. C is another point on the same meridian through A and its latitude is 35.4°S. Calculate, correct to 3 s.f., the angular difference or sum between:

1. A and B
2. A and C

Students’ activities: Find the angular difference or sum between A and C.

Solution

The angular sum between A and C (N and S) is θ = 34.6° + 35.4° = 70.0°.

How do you calculate the distance along a great circle?

Mode: whole class.

Teacher’s activities: The teacher leads the students to calculate the distance between points along the great circle.

In longitude and latitude, the earth is considered to be a sphere of radius 6400 km. The distance along a great circle is:

Distance = (θ / 360) × 2πR

• θ is the angular difference (or sum) in degrees.
• R is the radius of the earth, 6400 km.

Example 1

Two places on the same meridian have latitudes 23°S and 41°S. What is their distance apart measured along the meridian?

Example 2

Two places on the equator have longitudes 63°E and 132°E. How far apart are they measured along the equator?

Students’ activities: Two places on the equator have longitudes 132°E and 126°W. Calculate their distance apart.

How do you solve application problems on distance?

Mode: whole class.

Teacher’s activities: The teacher leads the students to solve application problems.

Example 1

Calculate the distance between A(24°N, 30°E) and B(24°S, 30°E) measured along the great circle. Give your answer correct to 3 s.f.

Example 2

Two places on the same line of longitude are 3949 km apart. If one of them is at latitude 18°N and the other is in the southern hemisphere, what is the latitude of the other?

Students’ activities: The difference between two latitudes on the same side of the equator is 15°. Find the distance between two points lying on a longitude that cuts across the latitudes, if they lie at the points of intersection between the longitude and the latitudes.

Evaluation

Mode: whole class.

The teacher asks the students to solve this question:

The distance between two places on the same longitude is 5302 km. Find the difference between their latitudes.

Assignment

Two points A(32°S, 30°E) and B(32°S, 12°W) are on the surface of the earth. P is another point on the same meridian as A, at latitude 28°N. Calculate the shortest distance between A and P on the earth, correct to 4 s.f. (Answer: 6700 km)

Reference materials

• MAN Mathematics for SS 3 (3rd edition)
• New General Mathematics for SS 3 by J. B. Channon et al.
• Multipurpose Mathematics for SSS by J. Olowofeso