The measurements are performed in three different places located in the USA, Mexico and Chile for each of the participating teams. Let’s call these places 1, 2 and 3. To get the result it will be convenient to consider three different pair of points (1 and 2), (2 and 3), and (3 and 1).

In what follows the procedure will be described for the (1 and 2) pair. Similar calculations should be performed with the remaining pairs and the final result for the Earth radius should be the average of the results obtained considering each of the pairs of points.

One of the purposes of this procedure is to determine the angle made by the vertical directions (as if the places were simultaneously located on the zenithal circle) at the two geographical points where the measurements were made. Let’s call it the “Vertical Angle”.

The vertical directions at different points on the Earth surface are not parallel because the Earth is almost spherical in shape. In this procedure the direction of the (parallel) sunlight rays is used as an aide to accomplish this goal.

The lengths of the gnomon and its shadow are used to determine the angle between the solar rays and a vertical line (as defined by a local plummet) at each place when the measurement (of the shortest shadow) is performed

Consider a simplified diagram of the gnomon and its shadow in two different geographical places as illustrated bellow

The red AC vertical line is the gnomon while the black horizontal AB line is its shortest shadow, in each of the places. The CAB angle is a right angle. We need to determine the ACB angle

Use graph paper. Construct a right angle with vertex at point A. Choose one of the angle sides to be horizontal so that the other one is vertical. On the vertical side determine point C by measuring (from A) a distance equal (or proportional) to the length of the gnomon. On the horizontal side determine point B by measuring (from A) a distance equal (or proportional) to the length of the shortest shadow. Now trace a line joining B and C.

VERY IMPORTANT The proportionality constants for both cases (gnomon and shortest shadow) MUST be the equal

To determine the Earth radius, the value of two such angles (at two different geographical points) are needed. If the two geographical points are located in the same hemisphere then the “Vertical Angle” is the subtraction between the angles determined at each of the geographical points. (see below). If the geographical points are located in different hemispheres then the “Vertical Angle” is the addition of the angles determined at each of the geographical points. (see below)

In the case of the contest which takes place in September and October 2007, if the pair of places where the measurements were performed are both located in the USA and/or in Mexico, the “Vertical Angle” is equal to the subtraction of both angles. If one place is located the USA or in Mexico while the other place is located in Chile the “Vertical Angle” is equal to the addition of both angles. If the places where the measurements were performed are both located in Chile, the “Vertical Angle” is the difference of both angles.

To add or subtract of the angles you may also do it by cutting the ABC triangles and perform the addition or subtraction by juxtaposing or superposing the ACB angles determined at both positions and then measuring the “Vertical Angle” with a protractor.

yields the ACB angles in each of the places. Trigonometric functions and their inverses may be computed with a scientific calculator (either an actual calculator or the one available in computers)

To determine the Earth radius, the value of two such angles (at two different geographical points) are needed. If the two geographical points are located in the same hemisphere then the “Vertical Angle” is equal to the subtraction between the angles determined at each of the geographical points. (see below). If the geographical points are located in different hemispheres then the “Vertical Angle” is equal to the addition of the angles determined at each of the geographical points. (see below)

In the case of the contest which takes place in September and October 2007, if the pair of places where the measurements were performed are both located in the USA and/or in Mexico, the “Vertical Angle” is the subtraction of both angles. If one place is located the USA or in Mexico while the other place is located in Chile the “Vertical Angle” is the addition of both angles. If the places where the measurements were performed are both located in Chile, the “Vertical Angle” is the difference of both angles.

To add or subtract the angles you may compute them separately and then proceed to add or subtract them. Nevertheless, it is more appropriate to do it by using a bit of trigonometry, once you have computed the tangent of the “Vertical Angles” (

) at each of the places, you may use

The angle between sunlight rays and the vertical at place 1 is equal to the ZO1 angle. In fact, the sunlight rays are parallel lines, the O1 line is a transversal line and alternate interior angles are equal, see next figure.

By the same token, the angles between sunlight rays and the vertical directions at place 2, at place 3 and at place 4 are equal to angle ZO2, angle ZO3 and angle ZO4, respectively.

To determine the Earth radius, the value of two such angles (at two different geographical points) are needed. If the two geographical points are located in the same hemisphere then the “Vertical Angle” is equal to the subtraction between the angles determined at each of the geographical points. (see below). If the geographical points are located in different hemispheres then the “Vertical Angle” is equal to the addition of the angles determined at each of the geographical points. (see below)

In the case of the contest which takes place in September and October 2007, if the pair of places where the measurements were performed are both located in the USA and/or in Mexico, the “Vertical Angle” is the subtraction of both angles. If one place is located the USA or in Mexico while the other place is located in Chile the “Vertical Angle” is the addition of both angles. If the places where the measurements were performed are both located in Chile, the “Vertical Angle” is the difference of both angles.

Insert the data requested (geographic latitude and longitude of both places and GMT of the measurements) and get the distance

Once the “Vertical Angle” 1O2 defined by places 1 and 2 (or any other couple of places) and the distance D12 between places 1 and 2 (or any other couple of places) as projected on the Meridian are known, we may establish that