Unit 11 Project
Indirect Measurements
By: David Astaburuaga, Humberto Torres, and Adrian Gutierrez
Class: Geometry 9 Honors
Period: 2
Date: 1/14/14
The Challenge
The assignment asked us to find the height of one object outside using indirect measurement. We used two methods
to determine the height of the outdoor object. One method is using a mirror and the other method is using the shadow
of the object you are trying to measure. First we did all the procedure necessary to find the object`s height, and we
later did the calculations. We decided to choose one of the red poles holding the nets behind the soccer goal as our
outdoor object.
How to Measure with the Two Methods
Shadow Method: To do this method we need to measure the shadow of the pole and a person`s shadow at the
same time of the day. Then we can measure the height of the person and do some calculations in order to get the
height of the pole, which is proportional to the person`s height. This is because both triangles are similar.
Mirror Method: To do this method you need to set a mirror on the floor and a person should position
him/herself in order to see the top of the object you are trying to measure in the middle of the mirror. Then you
should measure the distance from the person to the middle of the mirror and the object to the middle of the mirror
also. Then you should measure the person`s height up to his/her eyes. After this, you can calculate the height of the
object, because the height of the person is proportional to the height of the object. This is because both triangles are
similar.
Visual Representations
Here are some visual representation of the two types of indirect measurement methods we used to get the high of the
pole together with the measurements.
Mirror Method
In this representation, angle “a” is congruent to angle “b” because the line of vision from the top of the pole to the
mirror and from the person’s eyes to the mirror hits the mirror at the same angle. We can assume that both the pole
and the person are standing at a 90-degree angle making both angles also congruent.
Shadow Method
In this other representation angle “a” is congruent to angle “b” because the sun is hitting both the pole and the person
at the same angle. Again, we can assume that both the pole and the person are standing at a 90-degree angle, making
another pair of congruent angles.
The calculations
Now we will show and explain the calculations we did enable to find the pole’s height with both methods of indirect
measurement.
If you look back at the representations you will see that each has two pairs of congruent angles between both of the
triangles making them similar to each other by the similarity shortcut “AA”. Since both triangles are similar we were
able to use proportion to solve for the height of the pole. Here is how we solved the proportions:
1) Mirror Method Proportion
This shows how we found out the pole’s height using the mirror method. In equation at the top, the left ratio
is the distance from the mirror to both the pole and David. On the right ratio it shows the height of David and “x”
represents the pole’s unknown height. To solve it we did cross multiplication and after that we solve for “x”, so
we divided 4.83 by 0.76 to get 6.355m. So, because “x” is a variable for the height of the pole, 6.355m is it`s
height using the mirror method.
2) Shadow Method Proportion
This shows how we found the pole’s height using the shadow method. In the equation at the top, the left ratio
represents the length of the shadow in meters, for both David and the pole. On the right ratio it shows David’s
height, and “x” to represent the pole’s unknown height. So in order to solve this, we needed to do cross
multiplication, and that is just what we did. After that we solved for “x”, so we divided 14.889 by 2.27 to get
6.559m. This means that the height of the pole using the shadow method is 6.559m.
Conclusion and Problem Identification
After all of this we can now determine that the height of the pole using the mirror method is 6.355 meters and the
height using the shadow method is 6.559 meters.
As you probably noticed, the heights we got from both methods differs by about 30cm.We think that the variation
may have been caused due to the different grass heights, so when we were getting the measurements for the mirror
method, the mirror was not on a perfectly flat surface. If the mirror was inclined then the vision angles would not be
congruent because they would be hitting the mirror differently.
Frequently Asked Questions for Indirect Measurement
Mirror Method:
1) Should the height of the person or the distance to the eyes be used when measuring the person viewing the top
object in the mirror?
Answer: The height to the eyes should be used.
Explanation: The hypotenuse of the right triangle of the person looking to the mirror is actually the line of vision.
Since the line of vision starts at your eyes, because it is there where you see, the height that should be used is the
height to the person`s eyes.
2) Where on the mirror should you look for the top of the object?
Answer: You should look at the middle of the mirror.
Explanation: The middle of the mirror should be used as a marking point in order to be able to see the top and then
center it in the middle, because when you later measure it, you know you have to measure it to the middle of the
mirror.
3) When measuring the distances from the object to the mirror and the person to mirror, where should the
measurements start and stop?
Answer: For the person to the middle of his foot, and for the object directly bellow the highest point of the object.
And for both the middle of the mirror should be used to stop the measurement.
Explanation: The middle of your foot is about where your eyes are so that should be used because you are
measuring the height up to his/her eyes. An for the object, that is because the top you are seeing is the highest point
of the object so that is what you need to use if you want to measure your object up to it`s highest point. The middle
of the mirror should be used to stop the measurement because it is used as a reference point to later measure the
distance from the object to the mirror.
4) Does the mirror need to be flat on the ground or does it have to be slanted?
Answer: It has to be flat on the ground,
Explanation: If the mirror is slanted then the reflection angles will be different for the top of the object to your eyes
and from your eyes to the top of the object. If this was the case, then the two angles would not be congruent;
therefore you would not be able to prove both triangle similar because “AA”would not work.
Shadow Method:
1) Should the height of the person or the distance to the eyes be used when measuring the person and their shadow
that will be compared to the height/shadow of the object?
Answer: The total height of the person should be used.
Explanation: The highest point of a person`s head is actually the highest point on that person`s shadow so if you are
going to measure the shadow up to its highest point you should use the total height of a person since it is it`s
counterpart.
2) When measuring the shadow of the object and the shadow of the person, what should be the starting point for the
shadow?
Answer: For the person you should start measuring the shadow from the front of his/her feet and for the object you
should measure the shadow from the part that is directly opposite to the side facing the sun.
Explanation: You should start to measure the person`s shadow from the front part of its feet, and for the object you
should start to measure the shadow from the part that is directly opposite to the side facing the sun. This is because it
is the starting point of the shadow if you and the object are giving its back to the sun.
Division of Work
David Astaburuaga- Equations, and “The Calculations” portion. ______________________
Humberto Torres- “Visual Representations”and revision of essay. ______________________
Adrian Gutierrez- “The Challenge”,“Conclusion and Problem Identification”and“Summary”.
______________________