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  This lesson provided by:  
Author:Becky Cornelius
System: Lauderdale County
School: Lauderdale County Board Of Education

  General Lesson Information  
Lesson Plan ID: 26341


Geometric Mean and Indirect Measurement


In this inquiry-based lesson, students will work in groups to gather measurements needed to approximate the height of a building. Students will discover how geometric mean can be used to find the lengths of missing sides of right triangles. 

This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.

 Associated Standards and Objectives 
Content Standard(s):
MA2015 (9-12) Geometry
17. Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4]
MA2015 (9-12) Geometry
21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8]

Local/National Standards:

Alabama Course of Study 2009 standard is the same as the 2003 Course of Study

NCTM Standards addressed:

1.  In grades 9-12 all students should make decisions about units and scales that are appropriate for problem situations involving measurement.

2.  In grades 9-12 all students should make decisions about units and scales that are appropriate for problem situations involving measurement.


3.  In grades 9-12 all students should explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.



Primary Learning Objective(s):

Students will be able to use geometric mean to find the missing sides of right triangles.

Additional Learning Objective(s):

Students will be able to use a tape measure to find distances.

 Preparation Information 

Total Duration:

61 to 90 Minutes

Materials and Resources:

For each group: paper, tape measure, ruler, protractor

Technology Resources Needed:

Optional: LCD projector, calculators for students


Students should be familiar with how to read a tape measure to the nearest 1/8"

The teacher should locate an object outside that is too difficult to measure.  Possible objects are the top of a building, a flag pole, a light on the side of a building, etc.  The object should also have a flat ground surface to ensure accurate measurements.

Students should have knowledge of Pythagorean Theorem, special right triangles, ratios, proportions, similarity


Divide the class into heterogeneous groups of 3 or 4.  Groups should be organized in advance, with each group containing one high ability student, one low ability student, and one or two average ability student.

Possible student roles in groups: Recorder, Measurer, Paper Holder, Reporter 


1.  Ask all students to create a right triangle from a piece of paper.  They might draw the diagonal from corner to corner and cut the paper, or they might fold the paper to create a right triangle. 

2.  Once each student has a right triangle, have them number the angles 1, 2, and 3.  It is best to have all students number their angles the same for your discussion purposes.  One suggestion is to label the largest angle (the right angle) #3, the middle-sized angle #2, and the smallest angle #1.

3.  Have students accurately (with a protractor or other measuring device) draw the altitude of the right triangle.  This creates two smaller triangles. 

4.  Students should then assign a number (4, 5, 6, 7, 8, 9) to the three angles on each of the two smaller triangles.  Again, try to have as much consistency as possible in numbering.  Let the largest angle in the first triangle be #6, the middle angle #5, and the smallest angle #4.  Follow the same instructions for the second triangle.

5.  Now ask students to us their protractor to compare angles 4, 5, 6, 7, 8, and 9.

6.  Ask students questions such as:  "What do you notice about the angles?", "Which angles are the same?", "How do you know they are the same?" and "What does this tell us about the three triangles?"


1.  Now tell students that they will be using what they just discovered about similar right triangles to find the height of an object, such as a building, that would be difficult to measure. 

2.  Have students discuss in groups possible ways to find the height of an object too difficult to measure. 

3.  Ask for volunteers to share ideas.  If no group comes up with the idea of using a square or rectangle as discussed below, then suggest this as a possibility.  Allow students as much input as possible in planning the experiment, but guide them back to the triangles they created earlier.  They should see that the hypotenuse of the largest triangle represents the height of the building.  The altitude of the largest triangle represents the distance between their eyes and the building.  They will also need one other distance - the height from their eyes to the ground, which is represents a leg of one of the smaller triangles created earlier.

4.  Once outside, students need a piece of paper with a right angle.  Not all papers have an exact right angle, so to guarantee a right angle students should fold a piece of paper in half and then fold it in half again being sure to line up the edges of the paper.

5.  One student in the group faces the building and holds the corner of the paper (the vertex of the right angle) up to his eyes.  He gradually walks away from the building and stops when he is able to line up the top of the building with the top edge of the paper and the base of the building with the bottom edge of the paper.  (See attached Diagram of Experiment). 

6.  Another student now gathers the necessary measurements, the distance from the first student's eyes to the building and the height of the person's eyes.  The distance from eyes to building can be measured along the ground if necessary for accuracy.

7.  Now students should use similar triangles to create a proportion that can be used to find the missing lengths.  The length found initially will be the distance from the point where the first student looks horizontally at the building to the top of the building.  This distance will then need to be added to the height of the student's eyes to get the total height of the building. 


1.  Students should draw a diagram of their experiment and record the measurements on the diagram along with an explanation of how they arrived at their solution. 

2.  When back in the classroom, ask a few groups to present their findings to the class. Ask questions relative to the experiment, such as how the found certain measurements and how they set up their proportions.

3.  Introduce the term "geometric mean" and explain how it relates to their experiment.

4.  For an interactive demonstration of geometric mean, click the following link to access math warehouse online.

5.  Demonstrate how to find the geometric mean of two numbers by showing examples similar to the ones in the attachment Geometric Mean Problems.


1.  Have students work the following problem in their math journal as an extension to the lesson:

The length of the shorter leg of a 30-60-90 triangle is 20cm.  Find the length of the altitude to the hypotenuse.  Explain how you arrived at your answer. (The answer is 10 square root 3.)

2.  Allow students to share their answers and their explanations.

3.  Ask students how this problem relates to their prior knowledge of special right triangles.  Also ask them how geometric mean was used to find the solution.


1.  Wrap up the lesson by asking students to use Think-Pair-Share to summarize their learnings for the day.  They should think about what they learned for 1 minute.  Then pair with someone seated close to them to discuss what they learned.  Finally share their thoughts with the class.

2.  For practice provide students a copy of the attachment Geometric Mean Problems (or similar problems of your choice).

**Some files will display in a new window. Others will prompt you to download.


Assessment Strategies

Informal assessment occurs throughout the lesson by observing the students' experiments, asking questions relative to their learning, and listening to responses and/or reading journal responses to explanations. 

For formal assessment, use the attached Geometric Mean Problems above for sample assessment questions.


This lesson can be modified by increasing the level of difficulty of problems worked.


To assign students to groups, be sure to include a variety of ability levels in each group.  This will allow students who need extra assistance to have access to peer tutoring and peer coaching.  Following the lesson, students requiring further remediation will benefit from additional explanation from the teacher.  These students should also be allowed to work with the interactive lesson on Math Warehouse (link above).  Allowing these students the benefit of a visual representation of triangles and their geometric mean will help those having difficulty understanding the concepts.

Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

Presentation of Material Environment
Time Demands Materials
Attention Using Groups and Peers
Assisting the Reluctant Starter Dealing with Inappropriate Behavior

Be sure to check the student's IEP for specific accommodations.
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