Eratosthenes experiment

How do we measure the circumference of the Earth today?

In the age of modern technology, the answer to this question may seem trivial. After all, we have satellites and a GPS system, so measuring the circumference of the Earth is a breeze!

But in this case, is it possible to measure the circumference of our planet with an ordinary stick?

Yes of course! On March 22, 2021, the day after the vernal equinox, we did it! How? We used our knowledge of geometry!

For the measurement, we only needed: a gnomon, a tripod for its stabilization, a spirit level, a card stock base for a gnomon with drawn circles, a ruler / ruler (linear rule), and a sheet for calculations.

The measurement of the circumference of the Earth had to be taken during solar noon (i.e., above-mentioned), when the Sun is precisely southward above the horizon during the day. On different days of the year and in different places, the sun rises, towers and sets at different times, due to the difference between the solar time and the official time, and also due to the elliptical nature of the Earth's orbit.

We knew that on March 22, 2021, the solar noon for Siedlce was at 11:37:36.

So, let's start measuring!

First, we took our stick (rod) and measured its length. In our experiment, it was 80.6 cm long. Then we set it upright on the ground on which the card with drawn circles was lying, and we stabilized it in our tripod. Such a vertical rod positioned vertically, i.e. perpendicular to the Earth's surface, is called a gnomon.

We then measured the length of the gnome's shortest shadow at solar noon. We repeated the measurement several times and recorded these values.

The shadow length was 99.4 cm.

Using the trigonometric function "tangent" we calculated the angle of inclination between the gnomon and the hypotenuse of the right triangle that was created for us. From the arrays of the values of the tangent function for the function value known to us, we read the angle for which this function value is.

We can also do this as the inverse of the tangent function, or arcus tangent.

In our case, this angle was 50.9626 °.

Calculations, calculations ...

Having obtained the value of the angle in degrees, we had to calculate the angular difference between Siedlce and the sun point. The solar point is the point on the surface of the globe above which the sun is currently at its zenith.

The distance from Siedlce to the sun point on March 22 was 5703.74 km, and its location was 0 ° 50 'N.

The angular difference was 50.1293 °.

Then we put together a proportion to calculate the circumference of the Earth. The resulting equation was as follows: the determined angular difference divided by 360 degrees equals the distance between Siedlce and the sub-sun point (March 22, approximately lying on the equator), to the circumference of the Earth, which was unknown.

After the transformations, we obtained a ready pattern for the circumference of the globe. Now it was enough to just substitute the data and we have the result ready! We got the result of 40,961 km. This value is 886 km higher than the table value assumed for 40,075 km. The relative error in determining this value was equal to 2.27%, which is a satisfactory result.