For the Eratosthenes experiment at school students can measure the angle at the top using a protractor as shown in the image bellow.
Another way is using the trigonometric relation between the angle at top, the pole height and the shadow's length on the floor.
For this relation to be used accurately the pole must be perfectly vertical and the floor must be horizontal, to make pole heigth and shadow length perpendicular to each other. In that way the shadow's shape will be a right triangle.
In a right triangle tangent of an angle is the relation between the opposite side and the adjacent side given by the formula
The angle at top can therefore be calculated by the inverse function that is given by the formula