The Teaching of Geometry

by David Eugene Smith

Of the simple outdoor applications of the proposition, one of the best is illustrated in this figure.

To ascertain the height of a tree or of the school building, fold a piece of paper so as to make an angle of 45°. Then walk back from the tree until the top is seen at an angle of 45° with the ground (being therefore careful to have the base of the triangle level). Then the height AC will equal the base AB, since ABC is isosceles. A paper protractor may be used for the same purpose.

Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and standing pins at the vertices for the purpose of sighting.

To measure PC, stand at some convenient point A and sight along APC and also along AB. Then walk along AB until a point B is reached from which BC makes with BA an angle of the triangle (60°). Then AC = AB, and since AP can be measured, we can find PC.

Another simple method of measuring a distance AC across a stream is shown in this figure.

Measure the angle CAX, either in degrees, with a protractor, or by sighting along a piece of paper and marking down the angle. Then go along XA produced until a point B is reached from which BC makes with A an angle equal to half of angle CAX. Then it is easily shown that AB = AC.