Measuring tall objects using clinometer, shadow, and mirror method

TREASURE HUNT ON ESTIMATING TALL OBJECTS
BY: POTCHONG M. JACKARIA

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

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.

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

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/

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.

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

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

Finding the height of tamarind tree
tamarind tree with a shadow from a 3 feet
stick. Using proportion the height of the
flagpole (X) is given by:
ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑒𝑒
=
𝑠ℎ𝑎𝑑𝑜𝑤 𝑜𝑓 𝑡𝑟𝑒𝑒
𝑥
3𝑓𝑡
=
52.8𝑓𝑡
3𝑓𝑡
𝑥 = 52.8 𝑓𝑡

Finding the height of breadfruit tree
breadfruit tree with a shadow from a 3 feet
=
𝑥
3𝑓𝑡
=
59𝑓𝑡
5.3𝑓𝑡
𝑥 = 33.40 𝑓𝑡

Finding the height of coconut tree
coconut tree with a shadow from a 3 feet
=
𝑥
3𝑓𝑡
=
43𝑓𝑡
3.5𝑓𝑡
𝑥 ≈ 36.86 𝑓𝑡

Finding the height of flagpole
Using proportion the height of the
ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑜𝑙𝑒
𝐸𝑦𝑒 𝑙𝑒𝑣𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡
=
𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑝𝑜𝑙𝑒
𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟
𝑥
5.5𝑓𝑡
=
18𝑓𝑡
4𝑓𝑡
𝑥 ≈ 24.75 𝑓𝑡

tamarind tree (X) is given by:
=
𝑚𝑖𝑟𝑟𝑜𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑡𝑟𝑒𝑒
𝑥
5.5𝑓𝑡
=
29𝑓𝑡
3𝑓𝑡
𝑥 ≈ 53.17 𝑓𝑡

breadfruit tree (X) is given by:
=
𝑥
5.5𝑓𝑡
=
24.5𝑓𝑡
4𝑓𝑡
𝑥 ≈ 33.69 𝑓𝑡

coconut tree (X) is given by:
=
𝑥
5.5𝑓𝑡
=
20𝑓𝑡
3𝑓𝑡
𝑥 ≈ 36.67 𝑓𝑡

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

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.

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 =
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 ≈ 28 𝑓𝑡 + 5.5 ft ≈ 33.5 ft.

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 =
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 = 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.

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.

Measuring tall objects using clinometer, shadow, and mirror method

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