Measuring the Radius of the Earth

We can measure the radius of the Earth for ourselves, using nothing but a watch!

At sunset, measure the amount of time between which the Sun appears to set from the ground level and from the height of your head (h). When you are standing up, you can "see a little further around the corner" and the Sun takes a little bit longer to set. This picture shows the Earth and the Sun, looking down on the solar system from the top of the North Pole. The Earth is rotating counter-clockwise, so the Sun sets first at ground level and then later at head height. The radius of the Earth is identified as a distance R, and Symbol for the Greek letter Theta, used to represent an angle. is the angle the Earth rotates through during this time interval Symbol for the Greek letter Delta, used to represent the change in a value over an interval.T.

Physical setup for the experiment at sunset, showing a large circle for the Earth, and a small circle a great distance away for the Sun/sunlight that reaches the Earth. The sunlight follows a horizontal line from left to right, and the circle of the Earth's surface sits on top of this line. The Earth's radius is marked with a vertical line from its center down to the surface (where it is perpendicular to the sunlight line). A second radial line is drawn along the radius which is roughly ten degrees counterclockwise from the first, with the angle between them being labeled as Theta. Because the second line is slanted, the radial portion is labeled R and the remaining portion between the Earth's surface and the sunlight line (much less than R in length) is labeled h.
[NMSU, N. Vogt]

At sunrise, measure the amount of time between which the Sun appears to rise from the height of your head (h) and from the ground level. When you are standing up, you can "see a little further around the corner" and the Sun rises a bit sooner. This picture shows the Earth and the Sun, looking down on the solar system from the top of the North Pole. The Earth is rotating counter-clockwise, so the Sun rises first at head height and then later at ground level. The radius of the Earth is identified as a distance R, and the Greek letter Theta. is the angle the Earth rotates through during this time interval The Greek letter Delta.T.

Physical setup for the experiment at sunrise, showing a large circle for the Earth, and a small circle a great distance away for the Sun/sunlight that reaches the Earth. The sunlight follows a horizontal line from left to right, and the circle of the Earth's surface hangs below this line. The Earth's radius is marked with a vertical line from its center up to the surface (where it is perpendicular to the sunlight line). A second radial line is drawn along the radius which is roughly ten degrees clockwise from the first, with the angle between them being labeled as Theta. Because the second line is slanted, the radial portion is labeled R and the remaining portion between the Earth's surface and the sunlight line (much less than R in length) is labeled h.
[NMSU, N. Vogt]

Here is the relationship between your height h, the time interval The Greek letter Delta.T, and the radius of the Earth R, where R and h are measured in centimeters, The Greek letter Delta.T is measured in seconds, and The Greek letter Delta.T/240 is in units of degrees.

Equation for R, the radius of the Earth. R is equal to h times the cosine of ( Delta-T over 240 ) divided by the quantity [ 1 minus cosine of ( Delta-T over 240 ) ].

Here are the results for various intrepid members of a previous class who performed the experiment.

The Greek letter Delta.T (seconds) h (cm) R (108 cm)
451800.3
271630.8
141522.9
111474.7
121784.8
81609.4

People have measured the radius of the Earth in detailed ways and found it to be 6.4 × 108 centimeters. We did pretty well!