Thales used a method involving similar triangles to measure the height of pyramids. He observed that at a single point in time, the shadows of different objects form similar triangles. By comparing the ratio of his shadow's length to his height, Thales could determine the pyramid's height based on the ratio of its shadow's length at the same time. This technique works because corresponding sides of similar triangles have the same ratio. Though not perfectly accurate, it provided an early approximation of heights before modern precision methods.

1. Thales' Shadow How Thales Measured the Height of the Great Pyramids

2. Ideas How do you think the ancient Egyptians measured the height of the pyramids? Discuss the different ideas and why they may or may not work.

3. Similar Triangle Theorem Thales, an ancient philosopher, found a quick and easy way to measure the height of a pyramid. He used part of the similar triangle theorem, which states that a triangle with the same angles has proportionate sides. a a b b c c Similar Triangles: 1) 2)

4. Ratios of Similar Triangles Corresponding sides of similar triangles will have the same ratio. Expressed Mathematically: If (2 * a) = d Then (2 * b) = e And (2 * c) = f In other words, triangle (2) is twice as large as triangle (1). a d b c f Similar Triangles: (1) (2) e

5. Calculations If (3.4 * a) = d Then (? * b) = e And (? * c) = f ? = 3.4 Triangle (2) is 3.4 times larger than triangle (1) because all the sides are 3.4 times larger. a d b c f Similar Triangles: (1) (2) e

6. Calculations (continued) If d = (3.4 * a) Then e = (3.4 * b) d / e = (3.4 * a) / (3.4 * b) a d b c f Similar Triangles: 1) 2) e

7. Calculations (continued) (3.4 * a) / (3.4 * b) = (3.4 / 3.4) * (a / b) Note: (3.4 / 3.4) = 1 1 * (a / b) = (a / b) Therefore: a/b = d/e In other words, the sides of similar triangles are proportionate so their ratios will be equal.

8. Thales’ Shadow Thales realized that shadows of different objects at the same time of day produce similar triangles. He simply waited until the time of day when the length of his shadow equaled his height. l a a = l

9. The Pyramid’s Shadow Once he found the time of day where the length of his shadow equaled his height, he knew the pyramid’s shadow at the same time of day would equal its height. h = b

10. Modification Knowing what we know about similar triangles, did Thales have to wait for the time of day when the length of his shadow equaled his height? No In fact he could have measured the shadow’s at any time of day (as long as he measured both shadows at the same time) and calculated the pyramid’s height using the ratio of his height to his shadow’s length.

11. Example l a From the similar triangle theorem : a / l = h / b Thales was 6 feet tall, his shadow was 4 feet long, and the pyramid’s shadow had a base with side equal to 100 feet long at the same time of day. How high is the pyramid? Expressed Mathematically: 6 / 4 = ? / 100 The pyramid’s shadow is 25 times longer than Thales’ shadow because 100 ft. / 4 ft. is 25. According to the similar triangle theorem, if side b is 25 times longer than side l then side a must be 25 times longer than side h. So, 6 ft. x 25 = the height of the pyramid = 150 ft. Does 6/4 = 150/100? Yes

12. Accuracy of Shadows Measuring height from shadows isn’t exact, but it gives a very close approximation. Unless the measurements are taken at exactly the same time of day the sun will have moved and the shadows will have changed length. Uneven ground can also effect measurements. Today pyramids can be measured much more precisely using satellites, air pressure or density, and lasers.

13. Time For Fun! How tall is your school? Apply what you’ve learned to calculate the height of objects around your school!