Measuring tall objects using clinometer, shadow, and mirror method
- 1. TREASURE HUNT ON ESTIMATING TALL OBJECTS BY: POTCHONG M. JACKARIA
- 2. How the ancient Egyptians did calculated the height of the great pyramid? Measuring along the side of the pyramid will not give the real height of the pyramid but the slanting height. And it is not possible to measure directly from the top towards the bottom using a long rope since its made of a solid blocks. It is said that Thales, a teacher of Pythagoras, found the height of the Egyptian pyramids using a quick and easy way. Thales realized that shadows of different objects at the same time of day produce similar triangles. Then he waited for a time when his shadow equals his height therefore the shadow of the pyramid at that time must be equal to the actual height of the pyramid. Picture is from https://en.wikipedia.org/wiki/Thales_of_Miletus
- 3. But was it possible for Thales to measure the height of the great pyramid anytime of the day without waiting for the time when his height and shadow becomes equal? Yes. Because two triangles form by the shadow at the same time of the day must be similar. We say the rays are parallel. When using our height as reference, we can say the tree stands straight and so do you, so you and the tree are parallel. Finally, if the ground is flat, the tree's shadow and your shadow are parallel. The similar triangle theorem, states that a triangle with the same angles has proportionate sides. This is called the Shadow Method.
- 4. The Mirror Method It is done by placing the mirror on the ground between you and the object of unknown height. Next move forward or backward until you sight the top of the object in the middle of the mirror. Measure the distance between you and the center of the mirror and then the distance between the center of the mirror and the base of the object of unknown height. The reflection from the mirror causes our measurements (the smaller ones measuring from the mirror to the person's feet and the ground to the eyes) to form a triangle similar to the one formed by the mirror, the bottom of the object and the top of the object. Illustration from: http://www.cpalms.org/Public/PreviewResourceLesson/Preview/43087
- 5. The use of Clinometer A clinometer is an instrument used for measuring the angle or elevation of slopes. The theory behind the clinometer utilizes trigonometry. Trigonometry is frequently used for indirect measurements like measuring the height of an object. We can easily make our clinometer, you just need the following: Protractor, straw, tape, tread and weight. Picture from http://www.brighthubengineering.com/building-construction-design/46130-how-to-measure-a-tall-building-or-skyscraper-without-leaving-the-ground/
- 6. Read the angle shown, and subtract from 90° to find your angle of vision from your eye to the top of the pole .Record your results on your paper. Illustration is from https://www.monumentaltrees.com/en/content/measuringheight/ Look through the straw of your clinometer at the top of the light pole (or whatever object you're measuring). The weighted string should hang down freely, crossing the protractor portion of the clinometer.
- 7. We made a tour in our school and find the height of four objects using the three methods mentioned. We are interested to find the height of the following: Flag Pole Coconut Tree Breadfruit TreeTamarind Tree
- 8. USING THE SHADOW METHOD
- 9. Finding the height of Flagpole We compared the shadow form by the flagpole with a shadow from a 3 feet stick. Using proportion the height of the flagpole (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 = 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 𝑥 3𝑓𝑡 = 23𝑓𝑡 2.8𝑓𝑡 𝑥 = 24.64 𝑓𝑡
- 10. Finding the height of tamarind tree We compared the shadow form by the tamarind tree with a shadow from a 3 feet stick. Using proportion the height of the flagpole (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 = 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 𝑥 3𝑓𝑡 = 52.8𝑓𝑡 3𝑓𝑡 𝑥 = 52.8 𝑓𝑡
- 11. Finding the height of breadfruit tree We compared the shadow form by the breadfruit tree with a shadow from a 3 feet stick. Using proportion the height of the flagpole (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 = 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 𝑥 3𝑓𝑡 = 59𝑓𝑡 5.3𝑓𝑡 𝑥 = 33.40 𝑓𝑡
- 12. Finding the height of coconut tree We compared the shadow form by the coconut tree with a shadow from a 3 feet stick. Using proportion the height of the flagpole (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 = 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑠𝑡𝑖𝑐𝑘 𝑥 3𝑓𝑡 = 43𝑓𝑡 3.5𝑓𝑡 𝑥 ≈ 36.86 𝑓𝑡
- 13. USING THE MIRROR METHOD
- 14. Finding the height of flagpole Using proportion the height of the flagpole (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑜𝑙𝑒 𝐸𝑦𝑒 𝑙𝑒𝑣𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑝𝑜𝑙𝑒 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑥 5.5𝑓𝑡 = 18𝑓𝑡 4𝑓𝑡 𝑥 ≈ 24.75 𝑓𝑡
- 15. Finding the height of tamarind tree Using proportion the height of the tamarind tree (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝐸𝑦𝑒 𝑙𝑒𝑣𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑟𝑒𝑒 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑥 5.5𝑓𝑡 = 29𝑓𝑡 3𝑓𝑡 𝑥 ≈ 53.17 𝑓𝑡
- 16. Finding the height of breadfruit tree Using proportion the height of the breadfruit tree (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝐸𝑦𝑒 𝑙𝑒𝑣𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑟𝑒𝑒 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑥 5.5𝑓𝑡 = 24.5𝑓𝑡 4𝑓𝑡 𝑥 ≈ 33.69 𝑓𝑡
- 17. Finding the height of coconut tree Using proportion the height of the coconut tree (X) is given by: ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝐸𝑦𝑒 𝑙𝑒𝑣𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑟𝑒𝑒 𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑥 5.5𝑓𝑡 = 20𝑓𝑡 3𝑓𝑡 𝑥 ≈ 36.67 𝑓𝑡
- 19. Finding the height of flagpole This method uses trigonometry. Specifically we make use of tangent (although other trigonometric functions can be use). The initial height of the coconut tree (X) is given by: Tan ɵ = 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑜𝑙𝑒 𝑓𝑟𝑜𝑚 𝑒𝑦𝑒 𝑙𝑒𝑣𝑒𝑙) 𝑎𝑑𝑗𝑎𝑠𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑜𝑙𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟) Where ɵ is the angle of vision which is equal to 90°- the reading on the clinometer. Tan 45° = 𝑥 19𝑓𝑡 = 19𝑓𝑡 tan 45° = 19 ft Height of flagpole = x + height of observer (eye level) 𝑥 = 19𝑓𝑡 + 5.5𝑓𝑡 = 24.5 𝑓𝑡
- 20. Finding the height of tamarind tree This method uses trigonometry. Specifically we make use of tangent (although other trigonometric functions can be use). The initial height of the coconut tree (X) is given by: Tan ɵ = 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑓𝑟𝑜𝑚 𝑒𝑦𝑒 𝑙𝑒𝑣𝑒𝑙) 𝑎𝑑𝑗𝑎𝑠𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟) Where ɵ is the angle of vision which is equal to 90°- the reading on the clinometer. Tan 55° = 𝑥 33 𝑓𝑡 𝑥 = 33 𝑓𝑡 tan 45° ≈ 47.1288 𝑓𝑡 Actual height of tree = x + height of observer Actual height ≈ 47.1288 𝑓𝑡 + 5.5 ft ≈ 52.628 ft.
- 21. Finding the height of breadfruit tree This method uses trigonometry. Specifically we make use of tangent (although other trigonometric functions can be use). The initial height of the coconut tree (X) is given by: Tan ɵ = 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑓𝑟𝑜𝑚 𝑒𝑦𝑒 𝑙𝑒𝑣𝑒𝑙) 𝑎𝑑𝑗𝑎𝑠𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟) Where ɵ is the angle of vision which is equal to 90°- the reading on the clinometer. Tan 45° = 𝑥 28 𝑓𝑡 𝑥 = 28 𝑓𝑡 tan 45° = 28 ft Actual height of tree = x + height of observer Actual height ≈ 28 𝑓𝑡 + 5.5 ft ≈ 33.5 ft.
- 22. Finding the height of coconut tree This method uses trigonometry. Specifically we make use of tangent (although other trigonometric functions can be use). The initial height of the coconut tree (X) is given by: Tan ɵ = 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑓𝑟𝑜𝑚 𝑒𝑦𝑒 𝑙𝑒𝑣𝑒𝑙) 𝑎𝑑𝑗𝑎𝑠𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑟𝑒𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟) Where ɵ is the angle of vision which is equal to 90°- the reading on the clinometer. Tan 45° = 𝑥 31 𝑓𝑡 𝑥 = 31 𝑓𝑡 tan 45° = 31 ft Actual height of tree = x + height of observer Actual height = 31 𝑓𝑡 + 5.5 ft =36.5 ft. Note: Slight variations in the estimates using the three methods are due to the inaccuracy in the measuring devices used and in rounding of measures to foot.
- 23. PICTURES
- 26. Venue: Corazon Abubakar Iran Memorial Central Elementary School, Tandubas, Tawi-Tawi. Date: July 11-12, 2017 Materials: Meter stick, camera, clinometer Thanks to the following grade-VI pupils. Alchausar, Tahil, Sopian, Gary, and Albasil.