Stage: 2 and 3
Early man soon learned that joining forces with other people by living in communities was a way to keep safe. By pooling resources, such as their strength to do heavy jobs and their weapons, they could get more food, as well as wood for fires and for building, which they could share .
But to get
jobs done most effectively and to make sure that goods were
shared fairly, tools were needed. Some of the first mathematical
tools man developed where to help them measure. They had to
measure the passing of time, find ways to divide up pieces of
land, develop a system to help keep track of the number of
animals they owned and ways to measure grain so that they could
exchange goods with each other.
Another early way of measuring that was developed was to find
the heights of trees. People needed to know if the wood would
be large enough to use as strong beams for supporting a
building, or to make a boat, maybe even to act as a roller for
moving large stones for buildings.
Over time,
foresters, builders, surveyors, map makers and scientists have
created special tools to help them measure the height of trees,
buildings and other tall structures very accurately. One of the
tools is called a theodolite, the illustration shows a surveyor
using one. You may have seen one on a building site. It is still
important to measure tall trees - which are in fact the largest
and oldest living organisms. Did you know that the tallest tree
in the world is in Redwood National Park in California? It's as
tall as a 30 storey building! That's about 360 feet!!
Some of the early ways of measuring the height of tall structures are still in use today. They require very little equipment and can be used out in forests where special equipment is hard to come by. Why not try some of these methods for yourself? If you try more than one way you can then compare your results to discover if the answers you get are close - they should be. After all, the building or tree doesn't change its height because you change your measurement method! You might be surprised just how accurate you can be.
Native Americans had a very interesting and unusual way of seeing how high a tree was. They would bend over and look through their legs!
They would walk far enough from the tree to a find a place
where they were just able to see the top of the tree (from
their upside down position). The distance from this place to
the base of the tree was approximately the height of the tree.
Does it work? Actually it does and the reason is rather simple.
For a normal, fit and healthy adult (one who can bend over in
such a way), the angle that is formed as they look through
their legs is approximately 45 degrees. The angle between the
tree trunk and the ground is fairly close to 90 degrees and,
using what you know about the angles and sides of a triangle
you can work out the height of the tree. The height of the tree
and the distance from the tree to the person is about equal.
Therefore, knowing the distance to the tree gives you a good
idea about the height of the tree. It certainly saves having to
carry heavy surveying equipment around.
Loggers are people who cut down trees. They learned a great
deal from people who lived in forests for generations, like the
Native Americans. They might be cutting down trees that are
over a certain height and it is important that they have an
easy way to estimate which of the trees around them are to be
cut. You can try and estimate the height of a tree or a
building using their method.
You need a partner who stands at the base of the tree. Move a distance away from the tree then, holding your pencil at arms length, between your thumb and forefinger so that it brackets the height of your partner (the top and bottom of the pencil coincide with the top of the head and the bottom of the feet of your partner). Now, use this length to step out the height of the tree in "pencil lengths". You can now measure the height of your partner and multiply this by the number of pencils high the tree is. This is worth doing several times from different distances and working out an average to improve the accuracy.
This is quite easy, once you have the knack. Describing the method makes it sound way more complicated than it is. After a couple of tries, you'll be able to apply for a lumberjack's job!
Fancy trying a third method? Don't forget to be comparing the sizes that you are getting for the tree/building height to see how they match up.
You will need a partner again, and another thing that you will need is a sunny day. You might have to pick your time for this experiment.
We'll call this method Measuring Shadows. Wait until the tree casts a shadow then measure the length of the shadow. Stand next to the tree and have a friend or parent measure your shadow. Be sure to do it as soon as you can after you measure the tree shadow! Measure your height to find out how tall you are. Now for the good part - get those calculators out! You will need the 3 measurements:
- the length of the tree's shadow
- the length of your shadow
- your height
(Tree's Shadow / Your Shadow) x Your Height = Tree's Height
Can you see why this works? Why not have a go?
Famous mathematicians. Squares. Time. Measures - generally. Length/distance. History of measurement. Mixed trig ratios. Area. History of mathematics. Measuring equipment.
Published October 2004.
Copyright © 2007. University of Cambridge. All rights reserved.
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email: nrich@damtp.cam.ac.uk
