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# Students from three continents get it right by measuring the radius of the Earth

With a gnomon (vertical stick on a horizontal board), observing the projected shadow and performing simple calculations, the Greek mathematician and astronomer Eratosthenes of Cyrene (276-194 BC), who went on to direct the ancient Library of Alexandria, was able to measure the size of Earth quite accurately.

Following his scientific method, students from schools and institutes in Extremadura and the Canary Islands in Spain, Bahía Blanca in Argentina and New Caledonia (French island territory in the Pacific) achieved “surprising results, considering that it was an experiment carried out by school-age children”, he explains. Jesús Manuel Carballar Álvarez, professor at IES El Pomar in Jerez de los Caballeros (Badajoz), where the project was promoted.

Argentine students performing the experiment. / La Imaculada College (Bahía Blanca)

Bearing in mind that the Earth is flattened at the poles, and that the accepted or theoretical value the polar radius is 6356.8 km and the equatorial radius is 6378.1 km, the students got pretty close.

Young students from Spain, Argentina and New Caledonia have successfully measured the Earth’s dimensions as Eratosthenes did over 22 centuries ago

“By comparing the measurements two by two, we obtained some values ​​of the terrestrial radius: on the one hand, 6,365.08 km, which deviates 8.28 km from the theoretical value of the polar radius, that is, an error of only 0.1%; and 12.92 km from the equatorial radius, with an error of 0.2%”, highlights the professor.

“And on the other hand –he adds-, the value 6,387.07 km, with a difference in relation to the accepted value of the terrestrial equatorial radius of only 8.97 km, which means an error of 0.12%; and a slightly larger percentage in relation to the polar radius”.

The exercise was carried out on the 20th and 21st of March, during the spring equinox. At that moment the Sun is located just above the Earth’s equator and the angles that the students measured at certain times coincided with the latitude they were at.

Students from Tenerife (Canary Islands) carried out the exercise at the Teide Observatory. / IES La Orotava – Manuel González Pérez

“With simple instruments, the students verified that they live on a planet that has a (more or less) spherical shape, the evidence of which is in the different lengths of the shadows of the gnomons at solar noon in different places”, comments Carballar .

The professor also remembers that those who were in the northern hemisphere (Spain) confirmed that the shadows evolved from left to right, while those in the south (Argentina and New Caledonia) saw them move from right to left, a way of confirming what hemisphere of the earth do they live in?

### Computer programs

To take measurements and calculate the Earth’s radius, the students used the Stellarium and Google Earth programs, then shared the data in a spreadsheet, made videoconferences from three continents and, finally, this month of May, they presented the final results.

In addition, they investigated the figure of Eratosthenes and traced the route -also on Google Earth- of the first circumnavigation of the world carried out by Juan Sebastián el Cano, considered the apex and ultimate demonstration of terrestrial sphericity.

Earth radius measurement in New Caledonia. / Professional Secondary School François D’assise

“One of the innovative aspects was the adaptation of the calculation of the terrestrial radius to educational levels where the concept of tangent had not yet been explained, thanks to the use of Geogebra, a free software widely used in academia. mathematics,” says Carballar.

At the eratosthenes project The Institute of Mathematical Sciences (ICMAT), the Institute of Astrophysics of the Canary Islands (IAC), the National University of the South (Bahía Blanca, Argentina) and the current Library of Alexandria collaborated.

### Eratosthenes and the first measurement of the Earth’s radius

To estimate the Earth’s radius more than 22 centuries ago, Eratosthenes of Cyrene studied the shadows at two remote locations when the Sun reaches its highest point in the sky, i.e. at solar noon on the summer solstice.

He knew that at that time in the city of Siena (present-day Aswan, Egypt) the sun’s rays fell perpendicularly because he saw their reflection in a deep well in the city. However, in Alexandria, the obelisks and their staff stuck in the ground cast a small shadow.

From the inclinations of the shadows, he calculated the angle between the rays of the Sun and the vertical of Alexandria (7.2º) on the summer solstice and estimated the distance to Siena (about 5,000 stadia, just over 800 km).

By a simple rule of thumb (72º/360º = 800/x) he was able to calculate that the circumference of our planet should be around 39,750 km today. Taking into account that the circumference of the circle is 2πr, he obtained a value for the Earth’s radius of about 6,267 km, just one hundred kilometers less than the real one.

The Greek Eratosthenes measured the radius of the Earth in Egypt. /intef

### Participating centers in the Eratosthenes project

IES EL POMAR – Jerez de los Caballeros (Jesús Manuel Carballar Álvarez) IES ILDEFONSO SERRANO – Segura de León (María del Carmen Repilado Quintero) CEIP NTRA. MRS. DE GUADALUPE – Segura de León (Remedios Rodríguez Díaz) CEIP SAN FRANCISCO DE ASIS – Fregenal de la Sierra (Carmen Romero Regajo) IES CAROLINA CORONADO – Almendralejo (Maria Teresa Pereira) IES VIRGIN DE GRACIA – Oliva de la Frontera. (Juan José Palos Silva) IES JARANDA – Jarandilla de la Vera. (Victorian Collado Cuevas)

Canary Islands:

IES LA OROTAVA – Tenerife (Reyes Yaiza García Sánchez and Mr. Enrique Mesa)

New Caledonia (France)

FRANCOIS D’ASSISE PROFESSIONAL SECONDARY SCHOOL – Bourail (Pascal Gangutia)

Argentina:

LA IMMACULADA SCHOOL – Bahía Blanca, province of Buenos Aires (Sebastián Ferraro)