# Geometric solids nets: 3d Geometric Shapes — Nets

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## Lesson 14 | Geometry | 6th Grade Mathematics

### Objective

Describe features of and identify nets that match prisms and pyramids.

### Common Core Standards

#### Core Standards

The core standards covered in this lesson

• 6.G.A.4
— Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

### Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

1. Understand that a pyramid is a closed three-dimensional figure that has one polygon base with triangle faces extending from the base and meeting at a common vertex.
2. Understand that a prism is a closed three-dimensional figure that has two parallel, same-polygon bases that are connected to each other by rectangles.
3. Understand that a prism or pyramid can be “unfolded” and represented as a two-dimensional net.
4. Identify three-dimensional figures from their corresponding nets.

### Tips for Teachers

Suggestions for teachers to help them teach this lesson

SERP Poster Problems’ “Knowing Nets” includes a helpful video in Slide 1 of a box being opened into a net and then refolded.

#### Lesson Materials

• Template: Nets (1 per student)
• Three-dimensional solids (Teacher set) — We suggest purchasing a teacher set of folding nets for this lesson in order to model to students.

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### Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

#### Problem 1

A set of prisms and a set of pyramids are shown below.

a.   Compare and contrast the features you notice about prisms and pyramids. In particular, identify and compare the faces, edges, and vertices.

b.   Identify the base of each figure and name each figure.

c.   Describe how a pentagonal prism and a pentagonal pyramid are similar and different.

##### References

Open Up Resources Photo: Grade 6 Unit 1 Lesson 13

13.2: Prisms and Pyramids

#### Problem 2

Your teacher will demonstrate the flattening and folding of a prism and a pyramid using a net. Observe how a prism and pyramid can be flattened into a two-dimensional form called a net. Observe how a two-dimensional net can be folded up to re-form the three-dimensional polyhedron. Think about the shapes and features of the net that connect back to the three-dimensional figure.

Then, explore with your own nets using the nets provided in Nets. For each net, answer the following questions:

a.   Do you think this net will fold into a prism, a pyramid, or neither? Why or why not?

b.   If so, what is the name of the shape?

c.   Test out your answers by folding the net into a three-dimensional figure.

#### Problem 3

Match each net below to its corresponding three-dimensional figure. Name each three-dimensional figure.

### Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Problem Sets and Problem Set answer keys are available with a Fishtank Plus subscription.

A task that represents the peak thinking of the lesson — mastery will indicate whether or not objective was achieved

Determine if each claim is accurate. Explain why or why not.

a.   Claim: This net will fold into a square prism.

b.   Claim: This net will fold into a pentagonal pyramid.

##### Student Response

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Examples where students match nets to solids and solids to nets; ask students to name the solids
• SERP Poster Problems Knowing Nets

Handout #1 only; the other handouts will be used in upcoming lessons
• EngageNY Mathematics Grade 6 Mathematics > Module 5 > Topic D > Lesson 15

Problem Set
• GeoGebra Exploring Nets of Geometric Solids

#5: Which net matches the displayed solid? (Note, this requires computers and could be used as a class demonstration instead of as part of the problem set. It includes great visuals of three-dimensional shapes opening up to reveal the net.)

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Lesson 13

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Lesson 15

## Geometric Solids and why you need more than paper models

Posted on by Yana Mohanty

My friend from Spain recently invited me on a tour down mathematical memory lane. When she and her brother were in school, some time in the 1980’s, they made these geometric solids with their dad. Amazingly, their dad (who is now a granddad) kept the paper solids for all these years and recently gave them to my friend.

Geometric solids made of paper nets, circa 1985, Spain

I’ve always been curious about what different cultures find mathematically significant, so I was very eager to see the geometric solids. You can see how delightful the paper models are in the picture above. Notice the altitudes drawn on the faces of the tetrahedron, along with the markings for the right angles.

What’s most important is that my friend has positive memories of learning math due to her father’s thorough hands-on teaching methods, and is now teaching her own children math.

The models were starting to come apart at the seams after more than 20 years, and I could see that they were made from nets similar to this one:

Net of an icosahedron with tabs for gluing

I thought it might be fun to ask my friend’s 9 year old daughter to reconstruct some of these solids with Geometiles. We started with the hexagonal prism and then moved on to the icosahedron. You can see the girl examining her mom’s assembled paper solids in the pictures at the top.  As I watched the incoming 4th grader figure out how to make the paper models, I realized what an excellent starting point the paper models made. They gave us a tangible prompt for discussing the attributes of the shapes. For example, in the case of the icosahedron, one has to decide what shapes are needed for the faces, and how they are connected together.

Then a simple observation occurred to me: in the paper net, the triangles are already pre-assembled together. One need not necessarily make the observation that there are 5 triangles meeting at every vertex. This is a salient characteristic of the icosahedron; moreover, counting the number of polygons meeting at a vertex is a fundamental principle in the study of 3D geometry. However, in the paper net one simply has to fold along the creases and glue the tabs. When constructing an icosahedron out of Geometiles or any other construction set, one has to make a very deliberate choice of assembling 5 triangles at every vertex and applying this principle to every vertex until the figure is completed. This makes for a much more challenging and active learning scenario than using paper nets. Alternatively, you can construct your own net out of Geometiles, and then assemble it into an icosahedron.

Any way you do this, the paper models are an inexpensive and accessible starting point for the study of polyhedra. But to delve deeply into the subject, one needs to construct these polyhedra from completely unassembled parts.

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## Graphics mesh bodies and BREP mesh bodies — 2023

BREP mesh bodies and graphics mesh bodies are created from triangular polygons called facets. Each facet has three vertices and three edges, called edges.

Facets on a BREP mesh body can be assembled into faces. The resulting faces are identical to the faces of standard SOLIDWORKS BREP bodies, except that the mesh faces do not have a geometric description. For example, facets on a face of a mesh can create a rectangle, but the face is defined simply as a set of facets, not a rectangle with a height and width.

### Solid types generated from mesh

You can convert mesh files to three solid types:

• SOLIDWORKS Standard Solids BREP
• Mesh bodies BREP
• Graphic bodies

Standard BREP geometry can be added to create hybrid mesh models.

SOLIDWORKS BREP solids are standard solids used in the SOLIDWORKS software. They can be solid bodies or surface bodies. Does not include mesh surfaces. Each point on the face of these bodies can be determined using a mathematical equation.

BREP mesh bodies can be solid bodies or surface bodies. The bodies are made up of mesh facets. Facets can be grouped into faces. Facets can form a geometric shape, such as a rectangle, and a rectangle can have a BREP face associated with it, but not every point on every face can be determined using a mathematical equation. You can show or hide facet edges on BREP mesh bodies. In chapter
Tools > Options > System Options > Display
select or clear the Show facet edges in BREP mesh bodies check box. This includes bodies created with the Convert to Mesh Body tool. This also includes bodies imported from *.stl and *.3mf files using the Graphics Body import option, which you select and then convert to BREP mesh bodies.

This option does not affect the display of regular edges on BREP mesh bodies, which include edges created directly on import or created later with the Segment Imported Mesh Body tool.

Graphic bodies are defined entirely by their facets without reference to equations. For example, what looks like a circle in the graphics body is actually a collection of a huge number of triangles, some of whose edges approach the circle’s circumference point. These mesh edges do not actually create a mathematical curve.

Hybrid solid or surface bodies include mesh BREP geometry and standard SOLIDWORKS BREP geometry in the same model. Typically, hybrid mesh bodies are created by adding SOLIDWORKS features to BREP mesh bodies. Faces and edges created by SOLIDWORKS features contain as much of the standard BREP SW geometry as possible, rather than mesh BREP geometry.

### Geometric elements

Mesh body types can include geometric elements:

 Facets Triangular faces of each grid unit Faceted edges Edges or edges of mesh facets Facet tops Mesh facet tops Edges BREP Mesh Facets Edges Grid borders Tops BREP Vertices of the edges of the mesh faces BREP

The edges of BREP mesh bodies can approximate mathematical curves, but do not have any geometric attributes. Hybrid mesh bodies can contain a continuous face where one face region is BREP mesh geometry and the other face region is standard BREP geometry. A continuous edge is a single region made up of facet edges that form a BREP mesh edge and a standard BREP edge. These are not new geometry types, but rather combinations of existing geometry types in a hybrid mesh body.

## Composition of geometric bodies

Compositions of geometric bodies articulated with each other, which are otherwise called insets, are an important transitional step from the simplest geometric bodies to complex man-made and natural objects in the study of academic drawing. Without understanding the rules for embedding bodies into each other, depending on their design features, it is impossible to correctly and realistically draw buildings, cars and any other equipment, as well as plants or living beings, including humans.

The skills of drawing up ligaments with embedded bodies are especially necessary for applicants for creative and architectural universities in Moscow, since ligaments are one of the most common exam topics.

### Who needs to learn how to work on inserts and links and why

Links are volume-spatial compositions of varying complexity, which are composed of geometric bodies articulated with each other, that is, partially included in one another. A distinctive feature of such images is that they are not drawn from life, but created by imagination, for which it is necessary to thoroughly understand the design features of the bodies included in the bundle.

The main task when working on the drawing of a bundle is to clearly present the design of the bodies that make up the composition, and then depict the forms and their articulations, taking into account the perspective reduction, build complex falling shadows, work out a single and competent tone solution from the point of view of the laws of chiaroscuro.

Such a complex work cannot be done without basic artistic knowledge and a significant amount of previous practical work, first on geometric bodies, and then on settings.
Tasks for the implementation of bundles contribute to the training of skills of volumetric-spatial thinking.

A useful result of numerous exercises with embedded bodies will be the ability to mentally form a clear image of the finished work, an understanding of the design of sections and the features of refraction of faces in sections. In addition, in the course of work, one has to arrange shadows and reflexes, relying only on an understanding of the laws of chiaroscuro and one’s own imagination, which teaches one to operate with the knowledge gained at the previous stage of training, without reference to nature, and also develops spatial thinking.

The skills acquired during the study of the frames serve the draftsman well already at the stage of working on still lifes, and when depicting complex man-made and living objects, they are absolutely necessary. Compositions of this type are passed at the entrance exams by applicants from the Moscow Architectural Institute and other architectural universities, but the skills of compiling them are useful to everyone who is seriously involved in any activity, one way or another connected with fine art.

### Basic skills for working with embedded geometric bodies

When starting to work on embedded geometric bodies, it is necessary to understand well the properties of the main figures that determine the design features of three-dimensional bodies, as well as to confidently master and use the laws of geometric and aerial perspective and chiaroscuro. In addition, the initial skills of working not only with natural objects are required, but also some experience in depicting representation and imagination. And, of course, it is necessary to confidently master the main tool of the artist — a pencil, because all stages of work are carried out without the use of a ruler, compass and other auxiliary tools.

Moving on to such a complex task as creating embedded bodies is possible only after you have mastered the fundamental principles of working on individual geometric bodies and settings. If you are confident in building bodies taking into account position in space, perspective cuts and foreshortening, have mastered the work with chiaroscuro and tone, and possess the fundamental compositional principles and related practical skills, it’s time to start mastering ligaments.

### Tips for Composing Embedded Solids

Most often, a cube or a tetrahedron is taken as the basis for work, since it is most convenient to cut into them more complex geometric bodies — cones, cylinders, balls.

The use of a special grid helps to avoid errors in perspective constructions. The more numerous and varied the geometric bodies that you include in your composition, the more difficult it is to avoid distortions and mistakes — even if the base is built with high quality. The construction of the grid must be studied separately, as this is a rather difficult task that takes some time even for an experienced draftsman. However, if the grid is built correctly, it will save time later when working on the composition.

Precisely executed constructions are the basis of the image, on which the result directly depends. It is unacceptable to make a mistake at this stage, which is why, when studying embedded bodies, great importance is given to the methods of articulation of various bodies with each other.

There are links of varying degrees of complexity. The simplest are bodies with flat faces — cubes and tetrahedral prisms, and the study of the topic begins with them.

The visible part of the incised geometric body must allow to determine its dimensions, for which the visible parts must be half or more of its total volume. So, when articulated with a cylinder, it is necessary to show the circles of its bases and a significant part of the lateral surface, when articulated with a cone — the top, base and side surface.

A universal way to select the line along which the body will be embedded is to focus on the axes of symmetry, heights, and other lines and divisions inherent in the design of the bodies.

Bodies must collide with each other not less than one third and not more than half. If this principle is neglected, the compositions will either be too sparse, causing a feeling that the bodies barely touch each other, or so dense that it is already difficult to determine exactly which bodies they are composed of. Both cause a feeling of disharmony and are a violation of compositional laws, and therefore a mistake.

### How to arrange figures on a sheet

The correct arrangement of depicted objects on a sheet is the basis for successful work. If at the initial stage this was not paid enough attention or serious mistakes were made, the image will come out unconvincing.

Often, even for experienced draftsmen, the mistakes made become apparent closer to the middle or at the final stages of work, when it is already difficult to correct them. Practice, accuracy in creating constructions and grids, adherence to the logic of work, attention to detail, frequent assessment and self-checking for errors and inaccuracies will help to avoid this, for which it is useful to periodically move away from work at some distance.

When working on geometric bodies cut into each other, it will be necessary to mobilize not only the previously acquired skills of mental analysis of objects, their structure and sections, but also all available knowledge in the field of composition — in a word, to create to a large extent rationally, judiciously and consciously.

In fact, the choice of a compositional solution is a complex analytical work on the harmonious distribution of objects and masses in the plane of the sheet, which takes into account the tasks facing the artist in forming the emotional structure of the image and the chosen theme. This equally applies to both genre works and geometric abstractions.

#### The structure of the pictorial plane

The assessment of the blank sheet plane is determined by a number of features of human perception. The blank sheet plane already contains a conditional structure:
• horizontal and vertical axes defined by format edges;
• two diagonals given by opposite angles;
• the center, which is located at the intersection of conditional diagonal lines.

Composing in the imagination and thinking through the main parts of the future three-dimensional composition, we rely on this conditional structure. Subsequently, when we begin to work out the first sketches or draw up a grid, it will also become the basis on which we will form the structure of a real image.

#### Compositional balance

There are many ways to achieve the most important compositional task — harmony, but harmony will not work if the image is unbalanced. Depending on the artistic tasks, static or dynamic balance is used in the works. Static balance, as a rule, is built along vertical and horizontal axes, while static objects (for example, a cube or a prism) are more often used and are generally similar in shape, texture, and mass. To achieve a dynamic balance of the composition allows the placement of the main objects along the diagonal axes, the use of contrasts both in size, shape, texture, mass of objects, as well as in tone and color solutions.

When contemplating an image in which static balance is applied, the viewer’s attention is distributed over the entire plane, but if dynamic balance is applied, the viewer’s gaze, starting from the dominant, moves along the route specified by the artist, then comes to the geometric center and repeats the entire path several times.

Due to a number of anatomical and psychological features, we perceive objects and space with certain distortions that must be taken into account in order to achieve compositional balance.

Thus, we perceive the upper part of a sheet divided in the middle as a larger one, and the lower one as a smaller one, although we know that they are equal. Depicted exactly in the center, at the intersection of the diagonals, the object is visually shifted to the bottom of the sheet, which violates the compositional balance. That is why it is recommended to always place the compositional center of the image with some offset from the geometric center of the sheet. And depending on where it is shifted, the general character and mood of the whole work will change.

#### Composite center and secondary elements

Composite center is the dominant that is subordinated to and supported by other elements of the image. Although the weight and significance of secondary objects are different, each, even the smallest and most inconspicuous element, makes an equally important contribution to the formation of the composition as a whole. All elements are important in creating harmony, and each must be placed in its place and perform its own unique task, which cannot be “delegated” to other elements without losing a sense of overall integrity.

The most common ways to create a compositional center include: different size, shape, texture, color, position in the format. In addition, the compositional center requires the most careful study compared to the rest of the image.

### Examination requirements of the Moscow Architectural Institute for compositions of embedded bodies

The main requirements of the Moscow Architectural Institute for examination papers concern not only the quality of the construction of connections and the image as a whole, but also the exact observance of the conditions set by the university and are as follows:
• the examination drawing corresponds to the task proposed to applicants;
• high quality compositional ideas, the image is harmonious and its complexity is adequate to the task;
• the image is well arranged in the sheet;
• separate bodies are correctly built and depicted, perspective is taken into account, work with insets is correctly carried out;
• Proposed correct tonal solution;
• The image is complete.

In order to successfully pass the exam, the applicant must clearly understand the essence of the requirements of the university and demonstrate the knowledge and skills acquired during the preparation, an analytical approach, as well as the ability to create non-trivial works, manage fantasy and the creative process, working within a number of important restrictions.

### How to learn how to make links

Such a complex topic as embedded geometric bodies — especially taking into account the requirements of architectural institutes — cannot be mastered without a significant amount of practice under the guidance of an experienced mentor. Only by repeatedly working out and firmly fixing the understanding of the features of the main geometric shapes and the constructive structure of the bodies obtained on their basis, you will be able to compose and implement harmonious, compositionally thought-out bundles at a level worthy of entering the best architectural institutes of the capital.

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