Longitude and Latitude

Longitude and Latitudeh

Lesson Objectives

Understand the meaning of latitude and longitude.
Calculate angular distance between two points.
Compute distance along a meridian (longitude).
Compute distance along a parallel (latitude).
Use the formula for calculating distance along great circles and small circles.

Lesson Introduction

Latitude and Longitude are geographical coordinates used to specify the position of a point on the Earth's surface. This lesson explores how to measure distances using these coordinates including along meridians (great circles), parallels (small circles), and how to calculate angular distances.

Core Lesson Content

Latitude and Longitude Explained

Latitude: Distance north or south of the equator, measured in degrees (0° to 90°).
Longitude: Distance east or west of the Prime Meridian, also in degrees (0° to 180°).

Angular Distance

Angular distance is the angle between two points on the surface of the Earth measured at the Earth's center. It represents the shortest angle between their positions along a meridian (latitude) or a parallel (longitude).

Case 1: Same Hemisphere

When both points lie in the same hemisphere, subtract the smaller latitude from the larger:

$\text{Angular Distance} = | \theta_1 - \theta_2 |$

Example 1: Find the angular distance between A(30°N, 60°E) and B(70°N, 60°E).
Both points are in the Northern Hemisphere.

|70^\circ - 30^\circ| = 40^\circ

Case 2: Different Hemispheres

If the points lie in opposite hemispheres, add the absolute values of their latitudes:

$\text{Angular Distance} = |\theta_1| + |\theta_2|$

Example 2: Find the angular distance between A(25°N, 50°E) and B(35°S, 50°E).
Points are in different hemispheres.

25^\circ + 35^\circ = 60^\circ

Distance Along a Meridian (Great Circle)

Meridians are great circles. Use Earth’s radius $ R \approx 6400 \text{ km} $.

Formula:

$\text{Distance} = \frac{\theta}{360} \times 2\pi R$

Example 3: Find the distance between A(10°N, 25°E) and B(50°N, 25°E).

\theta = |50^\circ - 10^\circ| = 40^\circ

\text{Distance} = \frac{40}{360} \times 2\pi \times 6400 \approx 4469 \text{ km}

Distance Along a Parallel (Small Circle)

The distance depends on both the latitude and the difference in longitude.

Formula:

$\text{Distance} = \frac{\theta}{360} \times 2\pi R \cos(\phi)$

Example 4: Find the distance between A(45°N, 10°E) and B(45°N, 50°E).

\theta = |50^\circ - 10^\circ| = 40^\circ

\phi = 45^\circ

\text{Distance} = \frac{40}{360} \times 2\pi \times 6400 \times \cos(45^\circ) \approx 3157 \text{ km}

Great Circles vs Small Circles

Great circles: Largest circles through Earth's center (e.g., meridians).
Small circles: Circles not through the center (e.g., parallels).

Exercises

[WAEC] Find the angular distance between A(25°S, 40°E) and B(65°S, 40°E). [Past Question]
Find the angular distance between A(40°N, 55°W) and B(10°S, 55°W).
What is the angular distance between A(75°S, 80°W) and B(15°N, 80°W)?
Calculate the distance between A(10°N, 30°E) and B(50°N, 30°E).
[WAEC] Two places A(30°N, 80°W) and B(30°N, 20°W) lie on the same latitude. Find the distance. [Past Question]
[NECO] Calculate the great circle distance between A(0°, 5°E) and B(0°, 65°E). [Past Question]
[NECO] A(60°N, 10°E) and B(60°N, 70°E) lie on the same latitude. Find the distance. [Past Question]
Find the distance along the equator between A(0°, 15°E) and B(0°, 45°E).
Determine the distance along latitude 60°N between A(60°N, 20°W) and B(60°N, 80°W).
What is the angular distance between A(20°S, 60°W) and B(20°N, 60°W)?

Conclusion/Recap

By using latitude and longitude, we can compute angular distances and distances along both meridians and parallels. Great circles offer the shortest path between two points on a sphere, while small circles vary depending on latitude.