Math Detours · Grade 7 · Connections · 7

An address for every place on Earth

Two angles pin down any spot on the globe. How far north, how far round, and why a degree of longitude shrinks towards the poles.

Name ____________________Date ____________

Tonight's idea

The Earth is a ball, so a place needs two numbers, and both are angles.

Latitude says how far north or south you are. Picture standing at the centre of the Earth, facing the equator, then tilting up or down to look at the place. The equator is 0°, the North Pole is 90°N and the South Pole is 90°S.

Longitude says how far round you are, east or west of a line through Greenwich in London, from 0° to 180°. An address is written (latitude, longitude).

40° 40°N North Pole, 90°N equator, 0° Latitude: up from the equator 30° here: wide 30° at 60°N: half as wide Longitude: round from Greenwich
Two angles. The shaded slice covers the same 30° of longitude all the way up, but it gets narrower.

Once round the Earth through both poles is about 40,000 km, and a full circle is 360°. So one degree of latitude is \(40{,}000 \div 360\), about 111 km, everywhere.

Longitude is different. The lines of longitude all meet at the poles, so they squeeze together on the way there. Here is the length of one degree of longitude:

At latitude0°30°45°60°80°90°
km in 1° of longitude111967956190

Watch one: north, then east

A plane flies due north from 15°N to 60°N. That is 45° of latitude, and every degree of latitude is 111 km: \(45 \times 111 = 4{,}995\) km, about 5,000 km.

Then it flies due east along 60°N, from 10°E to 30°E. That is 20° of longitude. At 60° the table gives 56 km for each degree: \(20 \times 56 = 1{,}120\) km.

On the equator the same 20° would be \(20 \times 111 = 2{,}220\) km, about twice as far.

Your turn

  1. Two towns lie on the same line of longitude. How far apart are they if their latitudes are (a) 12°N and 47°N; (b) 8°S and 22°N?
  2. How long is a trip along 10° of longitude (a) on the equator; (b) at 45°N; (c) at 80°N?
  3. A ship sails once round the world, due east along 60°N. (a) How far does it sail? (b) How far is it once round the equator? Compare the two. (c) How far is it once round along 80°N?
  4. A plane starts at (0°, 0°). It flies north to 60°N, then east for 1,120 km, then south to the equator, then west for 1,120 km. (a) What is its longitude after the east leg? (b) How many degrees does the west leg cover? (c) Is the plane back where it started? If not, about how far away is it?

Stretch: the other side of the world

Your antipode is the point exactly opposite you, straight through the centre of the Earth. A city is at (40°N, 75°W). Find the address of its antipode. Then give a rule that works for any place.

Answers

  1. (a) \(47 - 12 = 35°\), and \(35 \times 111 = 3{,}885\) km. (b) From 8° south of the equator to 22° north of it is \(8 + 22 = 30°\), and \(30 \times 111 = 3{,}330\) km.
  2. (a) \(10 \times 111 = 1{,}110\) km. (b) \(10 \times 79 = 790\) km. (c) \(10 \times 19 = 190\) km.
  3. (a) \(360 \times 56 = 20{,}160\) km. (b) \(360 \times 111 = 39{,}960\) km, so the trip along 60°N is about half as long. (c) \(360 \times 19 = 6{,}840\) km.
  4. (a) At 60°N a degree of longitude is 56 km, and \(1{,}120 \div 56 = 20\), so it is at 20°E. (b) On the equator a degree is 111 km, and \(1{,}120 \div 111\) is about 10°. (c) No. It went 20° east and came back only about 10°, so it ends near (0°, 10°E), roughly \(10 \times 111 \approx 1{,}100\) km east of the start. The same distance covers more degrees near the pole.

Stretch. (40°S, 105°E). Rule: keep the latitude number and swap north for south. For longitude, go half way round: take the number away from 180 and swap east for west. Here \(180 - 75 = 105\).

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