Longitudes and Latitudes

Syllabus tag: KCSE | Mathematics | Form 4 | Topic 5 Longitudes and Latitudes

Lesson objectives

By the end of this topic, you should be able to:

• Define great and small circles in relation to a sphere, including the Earth.
• Establish the relationship between the radii of great and small circles.
• Locate places on the Earth using latitude and longitude.
• Calculate distances along great circles and along parallels of latitude, and solve problems on local time.

Longitudes and Latitudes

Position on the Earth is fixed by two angles, and distances between places follow from them.

a) Great and small circles

A great circle has its centre at the centre of the sphere. Its radius equals the Earth's radius R.

All meridians of longitude are great circles, and so is the equator.

A small circle has a smaller radius. Every parallel of latitude except the equator is a small circle.

b) The key relationship

A parallel at latitude θ has radius r = R cos θ.

It follows from a right-angled triangle. Its corners are the Earth's centre, the parallel's centre, and a point on the parallel.

At the equator θ is 0 and cos 0 is 1, giving r = R. At the poles θ is 90° and cos 90° is 0, giving r = 0.

Almost every calculation in this topic uses this relationship. Omitting the cos θ is the single most common error.

c) Locating a place

Latitude is measured north or south of the equator, from 0° to 90°.

Longitude is measured east or west of the Greenwich meridian, from 0° to 180°.

A position is written as (latitude, longitude), such as (1°S, 37°E) for Nairobi.

d) Distance along a meridian

Along a meridian, the radius is always R, so the distance depends only on the difference in latitude.

If both points are on the same side of the equator, subtract the latitudes. If on opposite sides, add them.

In nautical miles, distance = 60 × angle in degrees. One minute of arc along a great circle is one nautical mile by definition.

In kilometres, distance = (θ/360) × 2πR, with R about 6 370 km.

e) Distance along a parallel

Along a parallel of latitude θ, the radius is R cos θ, not R.

So distance = (difference in longitude ÷ 360) × 2πR cos θ.

In nautical miles, distance = 60 × longitude difference × cos θ.

Subtract the longitudes if both are east or both west. Add them if one is east and the other west.

f) Shortest distance

The shortest route between two places is along a great circle, not along a parallel.

That is why long flights curve towards the poles on a flat map. The parallel looks straight but is longer.

g) Local time

The Earth turns 360° in 24 hours, so 15° of longitude equals one hour.

Equivalently, 1° equals 4 minutes.

Places east are ahead in time; places west are behind.

Find the longitude difference, convert to time, then add going east or subtract going west.

Latitude has no effect on local time. Only longitude matters.

h) Where this is used

Navigation and flight planning. Time zone calculations. Satellite positioning. Shipping routes.

Words to know

• Great circle -- a circle on a sphere whose centre is the sphere's centre.
• Small circle -- a circle on a sphere with a smaller radius than the sphere.
• Meridian -- a line of constant longitude, running pole to pole.
• Parallel of latitude -- a line of constant latitude, running east-west.
• Nautical mile -- the distance subtending one minute of arc at the Earth's centre.

:::checkpoint Check yourself

1. What is the radius of a parallel at latitude 60°, in terms of R?
2. Find the distance in nautical miles from (20°N, 30°E) to (50°N, 30°E).
3. How many degrees of longitude correspond to one hour?
4. Why is the shortest route not along a parallel of latitude? :::

Bridge to practice

The exercises begin with great and small circles and locating positions, move through distances along meridians and parallels, and finish with nautical miles and local time. For every question, sketch the two positions first and decide whether the angles are added or subtracted.