Calculating Distances Using Longitude, Latitude & Great Circles – SS3 Mathematics Lesson Note (First Term)

The distance between two places on the Earth is found from the angular difference between them. Along a great circle, such as the equator, the distance is θ/360 × 2πR. Along a parallel of latitude, the formula is θ/360 × 2πR cos (latitude), with R = 6400 km and π = 22/7 where not given.

Behavioural objectives

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

1. Give the formula for finding distance along a great circle (the equator or a line of longitude).
2. Solve problems involving longitude and great circles.
3. Give the formula for finding distance along a parallel of latitude.
4. Solve problems involving latitude.

Instructional resources

A real globe and an orange.

How do we calculate distances using longitude and latitude?

The teacher presents the lesson with the steps below.

Step I: Identification of prior ideas

Mode: Entire class

Teacher’s activities: Displays the instructional resources and asks the students to identify the lines of longitude. The teacher then leads the students to give the formula for finding distance along a great circle.

Students’ activities: Students identify the lines of longitude and give the formula for finding distance along a great circle.

Lines of longitude

The formula for calculating distance along a great circle (the equator or a line of longitude) is:

Distance = θ/360 × 2πR

Step II: Problem solving on longitude and great circles

Mode: Individual

Teacher’s activities: Leads the students to solve problems involving longitude and great circles.

Students’ activities: Students solve problems involving longitude and great circles.

Example. Towns A and B lie on the equator. A has longitude 63°E, while B has longitude 123°E. Take the radius of the Earth as 6400 km.

1. What is the distance between the two towns along the equator?
2. How far is Santa from the North Pole?

Solution

(a) The angular difference between A and B = 123° − 63° = 60°.

Arc AB = 60/360 × 2πR = 60/360 × 2 × 22/7 × 6400 km = 6705 km (to the nearest km).

(b) The distance from the North Pole = arc AN. The angular difference is 90°.

Arc AN = 90/360 × 2πR = 90/360 × 2 × 22/7 × 6400 km = 10057 km.

Step III: Discussion on lines of latitude

Mode: Individual

Teacher’s activities: Displays the instructional resources and asks the students to identify the lines of latitude. The teacher then leads the students to give the formula for finding distance along a parallel of latitude.

Students’ activities: Students identify the lines of latitude and give the formula for finding distance along a parallel of latitude.

The formula for calculating distance along a parallel of latitude is:

Distance = θ/360 × 2πR cos (latitude)

Where θ is the angular difference (the difference in longitude) and the latitude is the angle of the parallel. Use R = 6400 km where it is not given, and π = 22/7.

Step IV: Application

Mode: Individual

Teacher’s activities: Leads the students to solve problems involving latitude.

Students’ activities: Students solve the problems in the evaluation questions below.

Closing the lesson

The teacher goes around to assess the students’ work and gives corrections on the board for the students to copy.

Evaluation

1. (Take the radius of the Earth as 6400 km and π as 22/7.)
   1. Find the longitude difference between A(06°N, 40°W) and B(60°N, 70°E).
   2. Find the distance measured along the parallel of latitude between A and B.
2. Calculate the difference in longitude between the following places:
   1. P(48°N, 25°E) and Q(48°N, 58°E).
   2. X(65°S, 40°E) and Y(65°S, 32°W).
3. Calculate the difference in latitude between the following places:
   1. P(49°N, 50°E) and Q(30°S, 50°E).
   2. X(60°N, 32°W) and Y(20°N, 32°W).

Assignment

Calculate the distances between the places given in parts (a) and (b) of questions 2 and 3 above.

References

1. Bakare L. Niyi, 2014, Mathematics Clinic.
2. Essential Mathematics for SS 3 students.