Connecting Geometry©

Chapter 9

Similar Triangles

Similar triangles can be very useful for measuring inaccessible objects. One method of doing this is called "shadow reckoning". To measure the height of a flagpole, for example, you would use the following procedure: measure the shadow of the flagpole,then hold a stick vertically nearby, and measure the shadow that the stick casts. As shown in the diagram below, the height of the flagpole is a vertical measurement and the stick is vertical, so we have a pair of right angles. The sun is a fixed point in the sky and so very far away that we can assume the rays are parallel; the sun's rays create equal angles at the top of the pole and the top of the stick. Therefore the two triangles are similar, by AA~, and we can write a proportion and find the needed height, x

Shadow reckoning was one of the great arts of the ancient Greeks, and was used extensively by early mathematicians, especially in measuring the heights of inaccessible objects. When the Greek mathematician Thales visited Egypt, he astonished the people with his use of shadow reckoning to find the height of the Great Pyramid. He used the same method as we do, but he had to make an adjustment in the method as shown below:

The beautiful image of the Pyramids below came from Nova's fascinating website devoted to Egyptology and the Pyramids. For more information about this interesting subject, click on the link below the picture:


Choose a tall object to measure, using indirect measurement. This object must be something that cannot simply be measured with a ruler or tape measure; it must be something that is very tall, and inaccessible, such as a tall flagpole at school, a high-rise building, or a steep cliff. It also needs to be in a relatively flat field, park, or section of town, so that its shadow will be measurable. This may depend on the time of day that you do your measuring, so you will probably need to make some observations in different parts of town, and at different times of the day. Of course the shadows will be shortest near noon!

Measure the length of your object's shadow, and measure the length of your own shadow as you stand nearby, or measure the length of the shadow of a yardstick held vertically nearby. If you use your own shadow, you may need an assistant, and you will need to know your own height.

Draw careful sketches, using a ruler and pencil (or the geometry software), and find the height of your object, using the method shown above. Write a careful explanation, including any problems you encountered. Tell us where and what your object is. Show your calculations. Be sure to use accurate values in your calculations, and do not round any numbers until you get your final answer. Presentation is always important; do a nice job on the drawings and the written work, so that it shows pride in your work.

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