Is the Earth round? Simple proofs
Long ago people saw clues that the Earth is round, like a ball.
- A ship sails away and its hull disappears first, then the mast. The sea curves.
- During a lunar eclipse the Earth's shadow on the Moon is always a circle edge.
- Travel north or south and new stars appear, old ones sink.
- Photos from space show a ball.
The Earth is not a perfect ball. It spins, so it bulges a little at the equator. The radius is about 6,371 km and the distance round is about 40,000 km. We call this shape an oblate sphere.
Eratosthenes measures the Earth
About 2,200 years ago a Greek scholar, Eratosthenes, heard that at noon in the town of Syene the Sun lit the bottom of a deep well. The Sun was straight overhead. The same day in Alexandria, far to the north, an upright stick had a shadow.
He measured the shadow angle: about 7.2 degrees. This is 1/50 of a full circle (360 degrees). So Alexandria is 1/50 of the way round the Earth from Syene.
The towns are about 800 km apart. So:
Circumference = 800 km x 50 = 40,000 km.
Why does it work? The Sun is so far that its rays are parallel. Two upright sticks point to the centre of the Earth. The angle between the sticks, at the centre, equals the shadow angle (like the alternate angles of parallel lines).
Triangulation and the length of a meridian
Later, surveyors needed a better way. A meridian is an imaginary line from the North Pole to the South Pole. To measure part of it they used triangulation.
- Measure one straight, flat distance very carefully. This is the baseline.
- From both ends, measure the angles to a far-away mark (a hilltop tower).
- Use the angles and the baseline to work out the other sides of the triangle (with trigonometry).
- Use the new side as a baseline for the next triangle. Build a chain of triangles.
Chains like this were used in the 1790s in France to measure the meridian. In India the Great Trigonometrical Survey measured a long arc from the south to the north. A long arc, plus the latitude of both ends, gives the size of the Earth. Measuring only angles is easy; measuring long distances is hard. Triangulation needs only one baseline.
Latitude and longitude
Latitude tells how far north or south of the equator a place is (0 to 90 degrees). Longitude tells how far east or west of the prime meridian it is (0 to 180 degrees). Together they give the address of any place. The full story of the grid is in the lesson Latitude and longitude.
Lines of latitude are parallel circles. Lines of longitude are half circles that meet at the poles.
Distances on a sphere
A straight line on a flat map is not the shortest path on a ball. The shortest path on a sphere is part of a great circle, a circle that cuts the Earth into two equal halves (the equator and all meridians are great circles). Aeroplanes fly along great circles.
Length of an arc = radius x angle (in radians), or distance = (angle / 360) x 40,000 km.
- 1 degree of latitude is always about 111 km.
- 1 degree of longitude is about 111 km at the equator, and gets smaller towards the poles: 111 x cos(latitude) km. It is 0 at the poles.
Try it: measure the Sun's angle with a stick
Push a straight stick upright in flat ground at noon. Mark the end of its shadow and measure the stick and the shadow. The shadow angle is found by drawing the triangle on paper and using a protractor. A friend in a town far north or south (at least 300 km away) does it the same minute. The difference of the two angles, and the distance between you, give the circumference with the formula above. Try it in the 3D too: move the distance slider and watch the answer stay near 40,000 km.
Key formulas and definitions
- Circumference = distance between towns x 360 / shadow angle
- Radius = circumference / (2 x pi)
- Arc length = (angle / 360) x circumference = R x angle (radians)
- 1 degree of latitude is about 111 km
- 1 degree of longitude is about 111 x cos(latitude) km
Worked examples
1. Two towns are 800 km apart. At the same moment the shadow angles differ by 7.2 degrees. Find the circumference of the Earth.
The angle is 7.2 / 360 = 1/50 of a circle. Circumference = 800 x 50 = 40,000 km.
2. The Earth's circumference is 40,000 km. Find its radius.
R = C / (2 x pi) = 40,000 / 6.28 = about 6,370 km.
3. How far apart are two places on the same meridian whose latitudes differ by 5 degrees?
1 degree is about 111 km. Distance = 5 x 111 = 555 km.
4. Two towns, 1,100 km apart, show shadow angles that differ by 10 degrees. What circumference does this give?
Circumference = 1,100 x 360 / 10 = 39,600 km.
5. A surveyor's baseline AB is 10 km. At A the angle to a tower C is 90 degrees, and at B it is 45 degrees. Find AC.
The triangle has angles 90, 45 and so 45 at C. So AC = AB = 10 km.
6. How long is one degree of longitude at 60 degrees north?
Length = 111 x cos 60 = 111 x 0.5 = about 55.5 km.
Common mistakes
- Thinking the shadow angle is measured on the ground. It is the angle between the stick and the Sun ray.
- Forgetting the Sun's rays are parallel. This only works because the Sun is very far.
- Using 111 km for one degree of longitude everywhere. It shrinks as you go towards the poles.
- Believing the shortest path between two places is the straight line on a flat map. On a sphere it is a great-circle arc.