### Summary

Students learn about the fundamental strength of different shapes, illustrating why structural engineers continue to use the triangle as the structural shape of choice. Examples from everyday life are introduced to show how this shape is consistently used for structural strength. Along with its associated activity, this lesson empowers students to explore the strength of trusses made with different triangular elements to evaluate the various structural properties.### Engineering Connection

Many engineers across various specialty fields engage in structural engineering, from aerospace engineers who design the satellite structures, to civil engineers who design bridges and highway flyovers, to mechanical engineers who design vehicle chassis and the placement of components inside computers and cell phones. The shapes included in these designs have significant effect on the strength of the structures. This lesson engages students in a discussion of the strength of various geometric shapes, such as squares and triangles, without the need for more advance physics-based analysis.

### Pre-Req Knowledge

Students should have a basic knowledge of geometry, specifically, know that regular geometric shapes such as squares, pentagons and hexagons can be reduced to triangles. They should also know how to calculate the sum of the interior angles of a polygon by reducing the polygon to triangles.

### Learning Objectives

After this lesson, students should be able to:

- List places where they see triangles used to increase structural strength.
- Explain why a triangle is the strongest geometric shape.

### More Curriculum Like This

**Triangles Everywhere: Sum of Angles in Polygons**

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**Polygons, Angles and Trusses, Oh My!**

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###
Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

###### International Technology and Engineering Educators Association: Technology

- Infrastructure is the underlying base or basic framework of a system. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### Colorado: Math

- Concepts of similarity are foundational to geometry and its applications. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### Colorado: Science

- Predict and evaluate the movement of an object by examining the forces applied to it (Grade 8) Details... View more aligned curriculum... Do you agree with this alignment?

###### National Council of Teachers of Mathematics: Math

- analyze properties and determine attributes of two- and three-dimensional objects (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

### Introduction/Motivation

Structural engineering is one of the oldest forms of engineering. The earliest buildings, roads, aqueducts and bridges all required structural design to make sure they were functional and safe. Structural engineering, though, is not unique to buildings. In fact, aerospace engineers use structural engineering when they design satellites, mechanical engineers use structural engineering when they design the frames of cars, even computer engineers use structural engineering to figure out how to best connect a video card to a motherboard!

In this lesson, you are going to learn about how structural engineers rely on fundamental geometries, with which we can easily predict performance, to design structurally sound objects and buildings.

(Next, show students the Strength of Shapes Presentation using the suggested script provided in the Lesson Background section.)

### Lesson Background and Concepts for Teachers

(The subsequent text aligns with the Strength of Shapes Presentation, a PowerPoint presentation. Make sure students have paper and pencil handy to sketch their ideas as they follow along with the presentation.)

(**Slide 1**) Today we will explore a fundamental structural engineering concept: the strength of shapes.

(**Slide 2**) When we look carefully at bridges, we can see how structural engineers use different shapes to make the overall design. We can see triangles and squares. We can even see parabolas.

(**Slide 3**) Structural engineers use the same types of shapes in buildings. Many building frames are simply repeating squares, as shown in the top left. The bottom left image shows how a square is reinforced by adding a diagonal cross brace in this scaffolding, which breaks the square into two triangles. The image on the right shows an Antarctic geodesic under construction. The structure of geodesic domes is similar to the structure of soccer balls and can be viewed as a group of pentagons and hexagons. But, if we break each of those shapes down, we can see that they are fundamentally composed of triangles.

(**Slide 4**) Even when we get outside the realm of civil or architectural engineering, we can see how engineers rely on the known strength of shapes. A motorcycle frame uses many triangles to support the wheels and seats. Mechanical engineers design cranes, which use triangles and squares in their frames. Even satellites use these familiar and basic regular geometries.

(**Slide 5**) On your paper, sketch each of these regular polygons: square, diamond and triangle. If we push straight down on a shape, putting the whole shape into compression, what happens to the shape? Draw, using a different pen or pencil or dashed line, how the shape would look if you pushed on it. Assume that the sides of the shape are rigid and won't change length or bend.

(**Slide 6**) Take a look at this! If you push down on top of the square, it will no longer be a square, but instead takes the shape of a rhombus, which is a type of parallelogram. This is called "racking." If we push down on the top of the diamond, it collapses down. But what about the triangle? The triangle maintains its shape!

(**Slide 7**) The reason that the square and diamond collapse is because the angle between the structural members can change without having the length of the members change or bend. Remember back to geometry when we talked about how polygons are defined? In this case, both quadrilaterals simply require the sum of the interior angles to equal 360 degrees, but each angle can change.

(**Slide 8**) Triangles are unique in that sense. The angle between two sides of the triangle is based on the length of the opposite side of the triangle. Do you remember this from geometry? The angle "a" is fixed, based on the relative length of side "A." Just like the angle "b" is fixed based on the relative length of "B" and "c" based on "C." This is why a triangle cannot collapse!

(**Slide 9**) As we showed, other regular polygons can be deformed without changing the length of the sides. A square loses its shape as its right angles collapse, and a pentagon and hexagon can be deformed. But the shapes stay "closed" because the sum of the interior angles is kept constant. For a shape with "n" sides, the sum of the interior angles will equal 180*(n-2). So a triangle's angles sum to 180 degrees, or 180*(3-2) degrees. A square's angles sum to 360 degrees, or 180*(4-2). So what can we do to the other shapes, the squares, pentagons and hexagons, to keep them from collapsing? Draw these shapes on your paper and add what would be necessary.

(**Slide 10**) Did you break the shapes into triangles? Since we know a triangle cannot collapse, and we know that these regular polygons can always be reduced to triangles (that's how we figure out the sum of the interior angles, remember?), breaking our polygons down into triangles keeps them from collapsing!

(**Slide 11**) The same concept applies in three dimensions. As shown, a cube can collapse by "racking," just like the square we saw collapse in two dimensions. So what would we do to make a strong 3D structure?

(**Slide 12**) We make 3D triangles! Specifically, we can make rectangular or triangular pyramids! This is why structural engineers rely on triangles, both in 2D and in 3D, to make strong structures! A 3D structure made of individual structural triangles like this is called a "truss," and is used throughout engineering for a strong light-weight structure!

### Vocabulary/Definitions

compression: A squishing force.

cross-brace: A diagonal structural member that breaks higher-order polygons down into simple triangles.

geodesic: A curved shape created by straight lines or objects.

parabola: The shape naturally formed by a rope held at both ends and allowed to sag; mathematically based on a quadratic equation.

racking: The process of a shape collapsing from a regular polygon into an irregular polygon.

regular polygon: A polygon that is equilateral (all sides of equal length) and equiangular (all interior angles are equal).

scaffolding: A temporary structure built around the outside of a building to give people safe access to high locations. For example to enable workers to lay brick, install trim or paint.

truss: A structure comprised of one or more triangular elements with straight individual members.

### Associated Activities

- Truss Destruction - Students construct trusses using Popsicle sticks and hot glue, and then test them to failure as they evaluate the relative strength of different truss configurations and construction styles.

### Lesson Closure

Now that we have reviewed the basics of how structural engineers rely on structural shapes, I expect that you will start to notice in your day-to-day life the way in which things are built. Look around you, at the buildings, and cranes, and bridges, and houses, and cars, and furniture, and you will see that so much of structural engineering is based upon these fundamental and simple shapes.

Next, in the activity, you will experience designing, building and testing structural trusses. So keep in mind the discussions we've had about the different shapes and how they can be used to make strong structures.

### Attachments

### Assessment

*Opening Question: *Ask students what regular geometry (triangle, square, circle, pentagon, hexagon, etc.) they think is the strongest and why. (Answer: A triangle is the strongest shape, and in this lesson, we will find out why!)

*Embedded Geometry Practice:* Have students participate by completing the two drawing tasks outlined in the PowerPoint presentation.

- Sketch each of these regular polygons on a sheet of paper: square, diamond and triangle. If we push straight down on the shape, putting the whole shape into compression, what happens to the shape? Draw, using a different pen or pencil or dashed line, how the shape would look if you pushed on it. Assume that the sides of the shape are rigid and won't change length or bend. (Answer: If you push down on top of the square, its 90 degree angles collapse and it becomes a simple rhombus, which is a type of parallelogram. If we push down on the top of the diamond, it collapses down. Nothing will happen to the triangle; it remains a triangle shape.)
- Sketch each of these polygons on a sheet of paper: square, pentagon and hexagon. What can we do to these shapes to keep them from collapsing? Draw these shapes on your paper and add what would be necessary. (Answer: Use lines to break the shapes into triangle shapes.)

*Personal Relevance:* After presenting the PowerPoint slides (or as a homework assignment), have students individually list on their papers the places, objects, structures and products where they have seen triangles functioning as structural shapes. After five minutes, have each student read his/her list to the class while you compile a master list of their responses on the board. (Possible answers: Bridges, transmission towers, cranes, peaked roofs, tables, chairs, bicycles, bike racks, railings, fences, gates, shelf supports, brackets, billboard sign supports, etc.)

### Lesson Extension Activities

Consider conducting the Polygons, Angles and Trusses, Oh My! lesson and its associated activities, Triangles Everywhere: Sum of Angles in Polygons and Polygons and Popsicle Trusses, during which students take a closer look at truss structures and draw polygon shapes to mentally “test” their forms and interior angles before/after load conditions are applied. During the activities, students divide regular polygons into triangles to calculate the sums of angles in polygons, and learn equations to find the sum of interior angles in a regular polygon and to find the measure of each angle in a regular n-gon. Then, to meet a hypothetical real-world challenge, they design, build and test strong and unique truss structures composed of Popsicle sticks and hot glue and then compare before/after polygon angle measurements to analyze the deformation of shapes

### Contributors

Darcie Chinnis, Amanda Guiliani, Scott Duckworth, Malinda Schaefer Zarske### Copyright

© 2013 by Regents of the University of Colorado### Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder### Acknowledgements

This digital library content was developed by the Integrated Teaching and Learning Program under National Science Foundation GK-12 grant no. DGE 0338326. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: April 26, 2017

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