The most accurate method for measuring the height of a flagpole is to use a surveyor's theodolite to measure the angle of elevation and the distance from the base of the pole. With that information, you can calculate the height from the tangent of the angle of elevation. You could accomplish the same thing with a rudimentary inclinometer made from a protractor and a weighted string, but there is a third method that yields a fairly accurate result with a lot less equipment and effort.
Erect the yardstick somewhere close to the flagpole so that the shadows from the flag pole and the yardstick are somewhat parallel. Use the carpenter's square or some 90-degree angle to ensure the yardstick is perpendicular.
Measure the length of the shadow of the yardstick and the flagpole. Since the ends of the shadows are in line with the tops of the respective objects and the sun, the lengths of their shadows are proportional.
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Calculate the ratio of flagpole shadow length (fs) to yardstick shadow length (ys). In this example, the lengths are measured in feet:
fs/ys = 33/3.3 = 10
Multiply the shadow ratio by the height of the yardstick (yh) to get the height of the flagpole. Since the measurements are in feet, the yardstick height is 3 feet.
yh * (fs/ys) = 3*10 = 30 feet
This is a great lesson for students who have not studied geometry or trigonometry. The relationship they are using is based on the tangent of the angle of elevation used in the more sophisticated methods.
This will only yield accurate results if the shadows are on level ground. A slope will distort the lengths. It is also best to use this method at least 1 hour before or after noon to have a shadow long enough to measure.