I was once taught a simple method for calculating distance using geometry. For example you can estimate the width of a river by selecting a point on the opposite bank. Can anybody out there help me remember how this is done?

You need an analogue watch for this.

Stand on the river bank facing directly opposite a known mark on the other bank - say that nice oak tree. Do not use sheep or cows as they tend to move about and ruin everything. Hold the watch horizontally before you so that the 12 points at the tree and the line joining the 9 and the 3 is parallel to your bank.

Now walk in the direction of the 3 (to your right), stopping every so often to take another bearing. Again hold the watch horizontally so that the 9 - 3 line is parallel to the bank and note where the oak tree is. You must walk until the oak tree is exactly at the 10:30 position on your watch.

The distance you walk is equal to the width of the river. You can count the paces and hence calculate the distance, as long as you know how long your pace is. But that's another question.

Note 1: By all means walk to your left if you prefer. You then need to get the tree to the 1:30 mark on your watch.

Note 2: If you don't have a watch - improvise. Draw one on some paper.

A method that I teach goes as follows:

1. Put a hat on.
2. Stand on the exact edge of the river.
3. Pull the hat down, obscuring your vision, until you can just see the edge of the opposite bank.
4. Without touching the hat, turn until you sight an object that you can walk to, at the edge of your vision.
5. Walk or measure off this distance--it's the same distance as the distance across the river.

This works by the principle of congruent triangles--the angle at which you're standing is 90 degrees, your height remains constant, and if you don't touch your hat, your angle of vision is constant. So by angle-side-angle, the triangles are congruent, and the distance you walk is the same as the distance across the river.

Here's a drawing, along with a legend about one of Napolean's officers using that method.

http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037878-X/Ch04/04-04/PH_Geom_ch04-04_Obj1.html

Wiki has a fairly advanced article on triangulation. The simple part is hidden down in the History section.

The use of triangles to estimate distances goes back to ancient times. In the 6th century BC the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids by measuring the length of their shadows at the moment when his own shadow was equal to his height; and to have estimated the distances to ships at sea as seen from a clifftop, by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff.

https://en.wikipedia.org/wiki/Triangulation

This site shows what we teach in middle school, using similar triangles:

Darnell, Angie, and Trevor are at a park along the Red Cedar River with their class.They decide to use similar triangles to find the distance across the river. After making several measurements, they sketch the diagram below.

http://www.aurora-schools.org/userfiles/273/Classes/12677/NESTED.TRIANGLE.METHOD.pdf