The development of the surface of the pyramid is a flat figure, composed of the base and faces of the pyramid, aligned with a certain plane. In the example below, we will consider the construction of a sweep using the triangle method.
Pyramid SABC is crossed by the frontally projecting plane α. It is necessary to build a development of the SABC surface and draw an intersection line on it.
On the frontal projection S""A""B""C"" we mark the points D"", E"" and F"", in which the trace α v intersects with segments A""S"", B""S"" and C""S"" respectively. Determine the position of the points D", E", F" and connect them to each other. The intersection line is indicated in the figure in red.
Determining the length of the ribs
To find the natural values of the lateral edges of the pyramid, we use the method of rotation around the projecting line. To do this, draw the i-axis through the vertex S perpendicular to the horizontal plane H. Rotating the segments SA, SB and SC around it, we move them to a position parallel to the frontal plane V.
The actual values of the edges are equal to the projections S""A"" 1 , S"" 1 B"" 1 and S""C"" 1 . We mark points on them D "" 1, E"" 1, F"" 1, as shown by the arrows in the figure above.
Triangle ABC at the base of the pyramid is parallel to the horizontal plane. It is displayed on it in full size, equal to ∆A"B"C".
Sweep construction order
In an arbitrary place on the drawing, mark the point S 0. We draw a straight line n through it and set aside the segment S 0 A 0 = S "" A "" 1.
We build the face ABS = A 0 B 0 S 0 as a triangle on three sides. To do this, from points S 0 and A 0 we draw arcs of circles with radii R 1 \u003d S "" B "" 1 and r 1 \u003d A "B", respectively. The intersection of these arcs determines the position of the point B 0 .
The faces B 0 S 0 C 0 and C 0 S 0 A 0 are constructed similarly. The base of the pyramid, depending on the layout of the drawing, is attached to any of the sides: A 0 B 0, B 0 C 0 or C 0 A 0.
Let us draw on the development a line along which the plane α intersects with the pyramid. To do this, on the edges S 0 A 0 , S 0 B 0 and S 0 С 0 we mark the points D 0 , E 0 and F 0 respectively. In this case, the point D 0 is located at the intersection of the segment S 0 A 0 with a circle of radius S""D"" 1 . Similarly, E 0 = S 0 B 0 ∩ S""E"" 1 , F 0 = S 0 C 0 ∩ S""F"" 1 .
GENERAL CONCEPTS ABOUT THE DEVELOPMENT OF SURFACES
We will consider the surface as flexible inextensible shell. In this case, some surfaces can be combined with the plane by transformation no breaks or wrinkles . Surfaces that allow such a transformation are called deployable.
The figure obtained by combining a developable surface with a plane is called a development.
The construction of developments is of great importance in the design of products from sheet material (vessels, pipelines, patterns, etc.).
Surfaces that develop geometrically accurate : polyhedral, conical, torsos, cylindrical.
Of the curved surfaces, the unfolding ones include those ruled surfaces (conical, cylindrical, torsos), in which the tangent plane touches the surface along its rectilinear generatrix.
All other curved surfaces are non-developing, but you can build them if necessary. approximate sweeps.
To build a development of any curved surface, it is divided into such curvilinear sections, each of which can be approximated by some flat figure, which requires to determine its nature only measurements.
For example:
The cylinder is divided into rectangles (Figure 16-1a);
a right cone into isosceles triangles (Figure 16-1b);
elliptical cylinder - into parallelograms (Figure 16-1c);
elliptical cone - into triangles (Figure 16-1d);
sphere - on a trapezoid.
PYRAMID AND CONICAL SURFACE REVEALS
As examples, consider the construction of developments of only four surfaces: a pyramid, a cone, a prism, and a cylinder.
Surface development of the pyramid
The development of such a surface is a flat figure, which is obtained by combining all its faces with one plane.
Example 1. Build a development of the surface of the pyramid ABCS (Figure 16-2) and draw a line MN on it .
Since the side faces of the pyramid are triangles, in order to build a development it is necessary to find the natural form of these triangles, for which it is necessary to determine the true lengths of the sides - the edges of the pyramid.
The base of the pyramid lies in a horizontal plane, therefore, the actual size of the ribs AB, BC and AC is already on the drawing.
Rib SA is a frontal, so it is depicted in front view in full size.
The nature of the ribs SВ and SC is determined by the method of a right-angled triangle. One of its legs is the excess of the point S over the points B and C, and the second is the top view of the ribs SB and SC.
Then, on three sides, we build successively all the side faces of the pyramid.
To plot the line MN on the development, we first determine the true value of the segments AM and B1 and put them on the development on the corresponding edges.
To plot the point M, we draw a straight line S2 on the face SBC and find its position on the development, setting aside the segment B2 (measured in the top view) on the side BC. Then, in the front view, draw a segment 3-4 through point 4 parallel to the edge BC and find its position on the development, for which we set aside the segment C4 on the side SC and draw a straight line 3-4 parallel to the edge BC through the resulting point. At the intersection of lines S -2 and 3-4 we find the point N. By connecting the obtained points M, 1, N we get the desired line.
Large selection of sweeps of simple geometric shapes.
Children's first exposure to paper modeling always starts with simple geometric shapes such as the cube and pyramid. Not many succeed in gluing a cube the first time, sometimes it takes several days to make a truly even and flawless cube. More complex cylinder and cone shapes require several times more effort than a simple cube. If you don’t know how to carefully glue geometric shapes, then it’s too early for you to take on complex models. Take care of yourself and teach your children to crate these “elements” of modeling from ready-made scans.
To begin with, I, of course, suggest learning how to glue an ordinary cube. Reamers are made for two cubes, large and small. A more complex figure is a small cube because it is more difficult to glue it than a large one.
So, let's begin! Download the development of all the figures on five sheets and print on thick paper. Before you print and glue geometric shapes, be sure to read the article on how to choose paper and how to cut, bend and glue paper in general.
For better printing, I advise you to use the AutoCAD program, and I give you a sweep for this program, and also read how to print from AutoCAD. Cut out the development of the cubes from the first sheet, be sure to draw a compass needle under the iron ruler along the fold lines so that the paper folds well. Now you can start gluing the cubes.
To save paper and just for every firefighter, I made several scans of a small cube, you never know if you want to glue more than one cube or something will not work the first time. Another simple figure is a pyramid, you will find its sweeps on the second sheet. Similar pyramids cost the ancient Egyptians, though not made of paper and not so small :)
And this is also a pyramid, only unlike the previous one, it has not four, but three sides.
Development of a trihedral pyramid on the first sheet for printing.
And another funny pyramid of five faces, its development on the 4th sheet in the form of an asterisk in two copies.
A more complex figure is the pentahedron, although the pentahedron is more difficult to draw than to glue.
Reamers of the pentahedron on the second sheet.
So we got to the complex figures. Now you have to tighten up, gluing such figures is not easy! To begin with, a regular cylinder, its development on the second sheet.
And this is a more complex figure compared to a cylinder, because at its base is not a circle, but an oval.
The development of this figure is on the second sheet, two spare parts were made for the oval base.
To accurately assemble the cylinder, its parts must be glued end-to-end. On the one hand, the bottom can be glued without problems, just put a pre-glued tube on the table, put a circle on the bottom and fill it with glue from the inside. Make sure that the diameter of the pipe and the round bottom fit snugly together, without gaps, otherwise the glue will leak and everything will stick to the table. The second circle will be more difficult to glue, so glue auxiliary rectangles inside at a paper thickness distance from the edge of the pipe. These rectangles will not let the base fall inward, now you can glue the circle on top without any problems.
A cylinder with an oval base can be glued in the same way as a regular cylinder, but it has a lower height, so it’s easier to insert a paper accordion inside, and put the second base on top and glue it along the edge.
Now a very complex figure - a cone. Its details are on the third sheet, a spare circle for the bottom on the 4th sheet. The whole difficulty of gluing the cone is in its sharp top, and then it will be very difficult to glue the bottom.
A complex and at the same time simple figure is a ball. The ball consists of 12 pentahedrons, the development of the ball is on the 4th sheet. First, the two halves of the ball are glued, and then both are glued together.
A rather interesting figure is a rhombus, its details are on the third sheet.
And now two very similar, but completely different figures, their difference is only in the base.
When you glue these two figures, you will not immediately understand what it is at all, they turned out to be some kind of completely unreceptive.
Another interesting figurine is the torus, only we have it very simplified, its details are on the 5th sheet.
And finally, the last figure from equilateral triangles, I don’t even know what to call it, but the figure looks like a star. Development of this figure on the fifth sheet.
That's all for today! I wish you success in this difficult work!
First, a development of an untruncated pyramid is built, all the faces of which, having the shape of a triangle, are the same. Point S 1 is marked on the plane (the top of the pyramid) and from it, as from the center, draw an arc of a circle with a radius R, equal to the actual length of the lateral edge of the pyramid. The actual length of the rib can be determined from the profile projection of the pyramid, for example, segments s" e" or s" b" , since these edges are parallel to the plane W and are depicted on it with a real length. Further along the arc of a circle from any point, for example but 1 lay six identical segments equal to the actual length of the side of the hexagon - the base of the pyramid. The actual length of the side of the base of the pyramid is obtained on a horizontal projection (segment ab). points a 1 - f 1 connect with a straight line to the top s 1 . Then from the top but 1 on these lines lay the actual lengths of the segments of the ribs to the cutting plane.
On the profile projection of a truncated pyramid, there are actual lengths of only two segments - s"5" and s"2". The actual lengths of the remaining segments are determined by rotating them around an axis perpendicular to the plane H and passing through the top s. For example, turning the segment s"6" around the axis to a position parallel to the plane W, we get its real length on this plane. For this, it is enough through the dot 6" draw a horizontal line until it intersects with the actual length of the edge SE (or SB). Section s // 6 0 // is the actual length of the segment S6 .
The resulting points l 1 , 2 1 , 3 1 etc. are connected by straight lines and the base and section figures are attached using the triangulation method. The fold lines on the scan are drawn with a dash-dotted line with two points.
Truncated Cone Reamer
The construction of a development of the cone surface begins with drawing a circular arc with a radius equal to the length of the generatrix of the cone from the point s 0 . The length of the arc is determined by the angle α:
α=
,
where d - diameter of the circumference of the base of the cone in mm;
l- the length of the generatrix of the cone in mm.
The arc is divided into 12 parts and the resulting points are connected to the vertex s about . From the top s 0 lay off the actual lengths of the segments forming from the top of the cone to the cutting plane R.
The actual lengths of these segments are found, as in the pyramid example, by rotating about a vertical axis passing through the top of the cone. So, for example, to get the actual length of the segment S2, it is necessary to draw a horizontal line from 2 "to the intersection at the point b / with the contour generatrix of the cone, which is its actual length.
To the development of the conical surface, the figures of the section and the base of the cone are attached.
Questions for self-examination
How to build a prism scan?
How to build a pyramid scheme?
How to build a cylinder scan?
How to build a sweep of a cone?
Theme: Axonometric Projections
Axonometric projections are a visual representation of an object on a plane, in which all three dimensions are depicted.
Axonometric projection is a parallel projection of an object along with a coordinate system onto a certain plane.
If the projecting beam is perpendicular to the projection plane, the axonometry is rectangular.
If not perpendicular - oblique.
The ratio of the length of the axonometric projection of the segment, // the axonometric axis, to its true length is the distortion coefficient.
k is the distortion coefficient along the OX axis
m is the distortion coefficient along the y-axis
n - distortion coefficient along the OZ axis
If k=m=n- axonometry is called isometry
If only two coefficients are equal (k=m≠n) - dimetria
Materials and fixtures
- paper or cardboard;
- scissors;
- pencil;
- ruler;
- compass;
- eraser;
- glue.
Defining parameters
Any geometric body has certain properties. This figure has a regular polygon base, and its vertex is projected into its center. An equilateral triangle is chosen as the base. This condition determines the name. The side edges of the pyramid are triangles, the number of which depends on the polyhedron chosen for the base. In this case, there will be three. It is also important to know the dimensions of all the constituent parts of which the pyramid will be composed. Paper sweeps are performed in accordance with all data of a geometric figure. The parameters of the future model are negotiated in advance.
How to make a pyramid out of cardboard?
The choice of material used depends on these data.
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Layout Assembly
Building a drawing
Completion of the simulation
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How to make a paper pyramid
Paper crafts begin with small boats and boats, with airplanes and pyramids that you can make with your child, instilling in him perseverance, creative imagination skills, introducing Japanese origami techniques that can develop logic and thinking. Some things in our house are iconic because they are made by hand, because they contain their own energy, they symbolize something for us. Initially, it was believed that the culture on the planet originated in Egypt, where the pyramid is still considered a symbol of energy conservation. It is called a sacred figure, which can contain a huge stream of charged particles inside, so many people make pyramids from ordinary paper, hollow inside, put knives or blades inside, which are a little dull, so that they become sharp and useful again in the household.
Mastering the technique of origami
- To make it, you need a sheet of paper 40/40 cm. First, we fold it from corner to corner, i.e. connect opposite sides.
Ways to make a paper pyramid
The craft is considered finished!
hollow pyramid
This method is very simple and fast. For a pyramid, inside which you can sharpen scissors, knives and blades, given the concentration of positive energy inside, you will need: a sheet of paper, PVA glue, scissors, a pencil
or marker.
On a sheet of paper, you need to draw a square and all parts of the pyramid adjacent to it, as shown in the drawing. This pattern needs to be cut only along the contour, all internal lines are fold lines that are marked with the blunt side of the scissors and bent inward. Do not forget that the pyramid will stick together, so leave a space of 1 cm for the substrate, which is then smeared with PVA glue and connected along the contour from all four sides.
The craft can be decorated with various Egyptian symbols (eye of Horus, ankh, rod, etc.) to attract even more energy inside.
Children's pyramid
Under strict parental guidance, children can also cope with the pyramid, let's try to take control of this process, you just need to look at the photo and bend the edges of the future product correctly.
- We outline the lines of the square diagonally: for this, we bend the opposite edges and unbend them.
Answers 100 to 1 answer who created the pyramid
Making a pyramid, how to build a pyramid correctly, how to make a paper pyramid, building pyramids.
How best to draw a pyramid
How to build a healing pyramid.
Pyramid energy - How to build a medical pyramid in the country with your own hands
Pyramid - development.
Development of geometric shapes
Rectangle, square, triangle, trapezium and others - geometric shapes from the exact science section. The pyramid is a polyhedron. The base of this figure is a polygon, and the side faces are triangles with a common vertex, or trapezoid. For a complete presentation and study of any geometric object, mock-ups are made. Use the most diverse material from which the pyramid is made. The surface of a polyhedral figure, developed on a plane, is called its development. The method of converting flat objects into volumetric polyhedra and certain knowledge from geometry will help to create a layout. It is not easy to make reamers from paper or cardboard. You will need the ability to perform drawings according to given dimensions.
Materials and fixtures
Modeling and execution of multifaceted three-dimensional geometric shapes is an interesting and exciting process. From paper, you can make a large number of various layouts. For work you will need:
- paper or cardboard;
- scissors;
- pencil;
- ruler;
- compass;
- eraser;
- glue.
Defining parameters
First of all, let's define what the pyramid will be. The development of this figure is the basis for the manufacture of a three-dimensional figure. Doing the job will require extreme precision. If the drawing is incorrect, it will be impossible to assemble a geometric figure. Suppose you need to make a layout of a regular triangular pyramid.
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How is a regular pyramid unfolded?
The basis of the model is a sheet of paper or cardboard. Work begins with a pyramid drawing. The figure is shown expanded. A flat image on paper corresponds to pre-selected dimensions and parameters. A regular pyramid has a regular polygon as its base, and its height passes through its center. Let's start with a simple model. In this case, it is a triangular pyramid. Determine the dimensions of the selected shape. To build a net of a pyramid, the base of which is a regular triangle, in the center of the sheet, using a ruler and a pencil, draw the base of the given dimensions. Next, to each of its sides, we draw the side faces of the pyramid - triangles. Now let's move on to building them. The dimensions of the sides of the triangles of the lateral surface are measured with a compass. We put the leg of the compass at the top of the drawn base and make a notch. We repeat the action, moving to the next point of the triangle. The intersection obtained as a result of such actions will determine the vertices of the side faces of the pyramid. We connect them to the base. We get a drawing of a pyramid. For gluing a three-dimensional figure, valves are provided on the sides of the side faces. We finish drawing small trapeziums.
Layout Assembly
Cut out the outline with scissors. Gently bend the scan along all lines. We fill the trapezoid valves inside the figure so that its faces close. Lubricate them with glue. After thirty minutes, the glue will dry. Volumetric figure is ready.
Development of a quadrangular pyramid
First, let's imagine what a geometric figure looks like, the layout of which we will make. The base of the chosen pyramid is a quadrilateral. Lateral ribs are triangles. For work, we use the same materials and fixtures as in the previous version. The drawing is done on paper with a pencil. In the center of the sheet, draw a quadrilateral with the selected parameters.
Divide each side of the base in half. We draw a perpendicular, which will be the height of the triangular face. With a compass solution equal to the length of the side face of the pyramid, we make notches on the perpendiculars, setting its leg to the top of the base. We connect both corners of one side of the base with the resulting point on the perpendicular. As a result, we get a square in the center of the drawing, on the faces of which triangles are drawn. To fix the model on the side faces, draw auxiliary valves. For reliable fastening, a centimeter-wide strip is enough. The pyramid is ready for assembly.
The final stage of the layout
The resulting pattern of the figure is cut out along the contour. Bend the paper along the drawn lines. The volumetric figure is collected by gluing. Lubricate the provided valves with glue and fix the resulting model.
Volumetric layouts of complex shapes
After completing a simple polyhedron model, you can move on to more complex geometric shapes. The development of a truncated pyramid is much more difficult to perform. Its bases are similar polyhedra. The side faces are trapezoids. The sequence of work will be the same as that in which a simple pyramid was made. The sweep will be more cumbersome. To complete the drawing, use a pencil, a compass and a ruler.
Building a drawing
The development of a truncated pyramid is performed in several stages. The side face of the truncated pyramid is a trapezoid, and the bases are similar polyhedra. Let's say they are squares. On a sheet of paper, we draw a trapezoid with the given dimensions. We extend the sides of the resulting figure to the intersection. The result is an isosceles triangle. We measure its side with a compass. On a separate sheet of paper we build a circle, the radius of which will be the measured distance.
The next stage is the construction of the side edges that the truncated pyramid has. The sweep is performed inside the drawn circle. The lower base of the trapezoid is measured with a compass. On the circle we mark five points that connect the lines with its center. We get four isosceles triangles. With a compass we measure the side of the trapezoid drawn on a separate sheet. This distance is set aside on each side of the drawn triangles. We connect the obtained points. The side faces of the trapezoid are ready. It remains only to draw the upper and lower bases of the pyramid. In this case, these are similar polyhedra - squares. Draw squares to the upper and lower bases of the first trapezoid. The drawing shows all the parts that the pyramid has. The sweep is almost ready. It remains only to finish the connecting valves on the sides of the smaller square and one of the faces of the trapezoid.
Completion of the simulation
Before gluing the three-dimensional figure, the drawing along the contour is cut out with scissors. Next, the scan is carefully bent along the drawn lines. Mounting valves are filled inside the model. Lubricate them with glue and press them to the edges of the pyramid. Let the models dry.
Making different models of polyhedrons
Making three-dimensional models of geometric shapes is an exciting experience. To master it thoroughly, you should start by performing the simplest scans. Gradually moving from simple crafts to more complex models, you can begin to create the most intricate designs.
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Making a pyramid, how to build a pyramid correctly, how to make a paper pyramid, building pyramids.
If you have not yet been imbued with the benefits of therapeutic pyramids.
If you still have doubts and are not convinced that you will be provided with truly unique and sought-after material on the topic: making a pyramid, how to build a pyramid correctly, building small or large pyramids with your own hands in the yard, in the country or at home, and also how to make a pyramid from paper.
Then read and watch more.
I hope this convinces you!
With the help of a pyramid, it is easy to get the so-called charged water, which is useful for washing with problematic skin, and drinking for diseases of the gastrointestinal tract (you can just for prevention). Creams and foodstuffs (such as oil) can be charged. One friend assures me that vodka that has stood overnight in the pyramid does not give a hangover, and the brine instantly relieves the hangover syndrome.
The pyramid is well treated for diseases that have a pain syndrome (you clearly feel how the pain is pulled out of you). With its help, you can normalize sleep and pressure, restore many functions of the body. Fractures grow together faster, postoperative sutures too. Enuresis and hemorrhoids are treated jokingly. The range of application of the pyramids is very wide, and has not yet been fully exhausted. Not all possibilities of medical pyramids are known.
The onset of otitis (inflammation of the middle ear) is cured in one session (thirty minutes), the emerging rotation of the nails disappears in two sessions (a total of three hours), all signs of food poisoning disappear ten minutes after you put the pyramid on your stomach etc. Everything is very simple if you manage to capture the very beginning of the disease. Then 1-2 sessions are enough for it to stop progressing and leave you alone.
It is more difficult with chronic ailments, however, many of them can also be cured with the help of a pyramid. It’s just that they need to be treated not so much during attacks (by the way, the pain during attacks with a pyramid is well extinguished), but to show patience and engage in treatment every day, say, watching TV. Eventually, the disease goes away.
So you can get rid of osteochondrosis in one and a half to two months. You can learn to stop attacks of congenital heart disease and angina pectoris at the very beginning, without waiting until the pain becomes severe.
The more often you use the healing pyramid, the more you are convinced that it saves your nerves, money and time. The treatment of any, even the smallest sore, with the help of today's healthcare system is a difficult, troublesome and expensive matter. And therefore, before running to the clinic, you can conduct an experiment and see if it is possible to cure it yourself?! And in the vast majority of cases, it is possible to do without doctors and pharmacies.
Do you want to know how the pyramid is made, how to build a pyramid, building pyramids with your own hands at home? Or, for example, how to make a paper pyramid? In general, it’s not difficult - you need to download and study my collection The right healing pyramid with your own hands at home.
Detailed instructions on how to properly build a pyramid are in the collection. Making a pyramid yourself according to a drawing using the instructions is not at all difficult, including making a paper pyramid.
What can you make a pyramid with your own hands at home? Any thin sheet material can be used except metals. These are plastic, plywood, hardboard, plexiglass, thick cardboard, you can even make a paper pyramid. You can even get by with ordinary drawing paper, but then the pyramid will turn out to be extremely short-lived.
P.S. This is only a small part of the reviews for this collection. Download it and you won't regret it. It will help not only cure incurable diseases, but also make your life easier. It will save a lot of time and money, and you will know how to build or make a pyramid with your own hands, or in extreme cases, how to make a paper pyramid.
You choose!
Do you want to learn more about the properties of the pyramids of the Golden Section (I offer you exactly such a pyramid), about their effect on living organisms and humans? About the effect of the pyramids on the structure of Space? On the effectiveness of pyramids in nuclear waste burial sites, in seismically active territories, in the territories of chemical and bacteriological waste burials, in territories with ozone layer anomalies, in territories with a difficult epidemiological situation, in solving the problems of drug addiction, alcoholism, in agriculture? Do you want to know about the experiments, conclusions, hypotheses of our scientists (in recent years, dozens of experiments with such pyramids have been carried out at the Research Institute of the Russian Academy of Sciences, RAMS)?
Then go to the information site www.carole2.narod.ru - everything is described in detail there. In order to save your traffic, you can download my zip-archive (148 Kb) and then read it. This archive contains the contents of the above site.
Look at some screenshots from instructions for making a pyramid, how to build a pyramid correctly, how to make a paper pyramid, building pyramids.
Of course, I will not give explanations here.
All detailed explanations on how to make a pyramid, and a much larger number of diagrams and drawings are in the collection itself. Moreover, you do not need to be a mathematician, or have a profession of a builder. Everything is shown and told, as they say "on the fingers". You need to download the collection The correct medical pyramid with your own hands at home, and study.
You just look at a small fraction of the drawings presented here, and decide for yourself whether the information you need is in this collection?
Personally, I have no doubt!
In addition to the technique of how to build a pyramid correctly or how to make a paper pyramid with your own hands, you will be given practical advice on using the pyramid and how to arrange it.
Well? Did I manage to convince you?
Hope so.
And I want to add a very important detail:
My collection "The Correct Do-It-Yourself Healing Pyramid" includes not only practice on how to properly build a do-it-yourself healing pyramid, but also theory. As you know, without theory there is no practice. And before starting the construction of a medical pyramid, you need to know at least a little theories about the subject, why it is so, and not otherwise, and what comes from.
In the collection "The correct medical pyramid with my own hands" I included:
– forms and properties of pyramids, use, methods of treatment;
- pyramids of precious and semi-precious stones;
- shungite pyramids;
– geometry of the pyramids of the Golden Section – full description, theory,
formulas, calculating sizes to make a pyramid with your own hands;
- how to make a pyramid out of paper, 50 cm in size. - the correct medical pyramid of the Golden Section (an example is given of how to make a pyramid out of paper, but you can, of course, make a pyramid out of other materials);
- book by Olga Sukhova.
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Pyramid - development. Development of the pyramid for gluing. Paper reamers
Rectangle, square, triangle, trapezium and others - geometric shapes from the exact science section. The pyramid is a polyhedron. The base of this figure is a polygon, and the side faces are triangles with a common vertex, or trapezoid. For a complete presentation and study of any geometric object, mock-ups are made. Use the most diverse material from which the pyramid is made. The surface of a polyhedral figure, developed on a plane, is called its development. The method of converting flat objects into volumetric polyhedra and certain knowledge from geometry will help to create a layout. It is not easy to make reamers from paper or cardboard. You will need the ability to perform drawings according to given dimensions.
Materials and fixtures
Modeling and execution of multifaceted three-dimensional geometric shapes is an interesting and exciting process. From paper, you can make a large number of various layouts. For work you will need:
- paper or cardboard;
- scissors;
- pencil;
- ruler;
- compass;
- eraser;
- glue.
Defining parameters
First of all, let's define what the pyramid will be. The development of this figure is the basis for the manufacture of a three-dimensional figure. Doing the job will require extreme precision. If the drawing is incorrect, it will be impossible to assemble a geometric figure. Suppose you need to make a layout of a regular triangular pyramid.
Any geometric body has certain properties. This figure has a regular polygon base, and its vertex is projected into its center. An equilateral triangle is chosen as the base. This condition determines the name. The side edges of the pyramid are triangles, the number of which depends on the polyhedron chosen for the base. In this case, there will be three. It is also important to know the dimensions of all the constituent parts of which the pyramid will be composed. Paper sweeps are performed in accordance with all data of a geometric figure. The parameters of the future model are negotiated in advance. The choice of material used depends on these data.
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How is a regular pyramid unfolded?
The basis of the model is a sheet of paper or cardboard. Work begins with a pyramid drawing. The figure is shown expanded. A flat image on paper corresponds to pre-selected dimensions and parameters. A regular pyramid has a regular polygon as its base, and its height passes through its center. Let's start with a simple model. In this case, it is a triangular pyramid. Determine the dimensions of the selected shape. To build a net of a pyramid, the base of which is a regular triangle, in the center of the sheet, using a ruler and a pencil, draw the base of the given dimensions. Next, to each of its sides, we draw the side faces of the pyramid - triangles. Now let's move on to building them. The dimensions of the sides of the triangles of the lateral surface are measured with a compass. We put the leg of the compass at the top of the drawn base and make a notch. We repeat the action, moving to the next point of the triangle. The intersection obtained as a result of such actions will determine the vertices of the side faces of the pyramid. We connect them to the base. We get a drawing of a pyramid. For gluing a three-dimensional figure, valves are provided on the sides of the side faces. We finish drawing small trapeziums.
Layout Assembly
Cut out the outline with scissors. Gently bend the scan along all lines. We fill the trapezoid valves inside the figure so that its faces close. Lubricate them with glue. After thirty minutes, the glue will dry. Volumetric figure is ready.
Development of a quadrangular pyramid
First, let's imagine what a geometric figure looks like, the layout of which we will make. The base of the chosen pyramid is a quadrilateral. Lateral ribs are triangles. For work, we use the same materials and fixtures as in the previous version. The drawing is done on paper with a pencil. In the center of the sheet, draw a quadrilateral with the selected parameters.
Divide each side of the base in half. We draw a perpendicular, which will be the height of the triangular face. With a compass solution equal to the length of the side face of the pyramid, we make notches on the perpendiculars, setting its leg to the top of the base. We connect both corners of one side of the base with the resulting point on the perpendicular. As a result, we get a square in the center of the drawing, on the faces of which triangles are drawn. To fix the model on the side faces, draw auxiliary valves. For reliable fastening, a centimeter-wide strip is enough. The pyramid is ready for assembly.
The final stage of the layout
The resulting pattern of the figure is cut out along the contour. Bend the paper along the drawn lines. The volumetric figure is collected by gluing. Lubricate the provided valves with glue and fix the resulting model.
Volumetric layouts of complex shapes
After completing a simple polyhedron model, you can move on to more complex geometric shapes. The development of a truncated pyramid is much more difficult to perform. Its bases are similar polyhedra. The side faces are trapezoids. The sequence of work will be the same as that in which a simple pyramid was made. The sweep will be more cumbersome. To complete the drawing, use a pencil, a compass and a ruler.
Building a drawing
The development of a truncated pyramid is performed in several stages. The side face of the truncated pyramid is a trapezoid, and the bases are similar polyhedra. Let's say they are squares. On a sheet of paper, we draw a trapezoid with the given dimensions. We extend the sides of the resulting figure to the intersection. The result is an isosceles triangle. We measure its side with a compass. On a separate sheet of paper we build a circle, the radius of which will be the measured distance.
The next stage is the construction of the side edges that the truncated pyramid has. The sweep is performed inside the drawn circle. The lower base of the trapezoid is measured with a compass. On the circle we mark five points that connect the lines with its center. We get four isosceles triangles. With a compass we measure the side of the trapezoid drawn on a separate sheet. This distance is set aside on each side of the drawn triangles. We connect the obtained points. The side faces of the trapezoid are ready. It remains only to draw the upper and lower bases of the pyramid. In this case, these are similar polyhedra - squares. Draw squares to the upper and lower bases of the first trapezoid. The drawing shows all the parts that the pyramid has. The sweep is almost ready. It remains only to finish the connecting valves on the sides of the smaller square and one of the faces of the trapezoid.
Completion of the simulation
Before gluing the three-dimensional figure, the drawing along the contour is cut out with scissors. Next, the scan is carefully bent along the drawn lines. Mounting valves are filled inside the model. Lubricate them with glue and press them to the edges of the pyramid. Let the models dry.
Making different models of polyhedrons
Making three-dimensional models of geometric shapes is an exciting experience. To master it thoroughly, you should start by performing the simplest scans. Gradually moving from simple crafts to more complex models, you can begin to create the most intricate designs.
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If you need step-by-step instructions on how to build a pyramid net, then I ask you to our lesson. First of all, evaluate whether your pyramid is unfolded in the same way as in Figure 1.
If you have it turned at 90 degrees, then the edge marked in the figure as "known real values" in your case can be found on the profile projection, which you will need to build. In my case, this is not required, we already have all the quantities necessary for constructing. It is important not to forget that in this drawing only the edges SA and SD on the frontal projection are displayed in full size. All others are projected with length distortion. In addition, in the top view, all sides of the hexagon are also projected in full size. Based on this, let's get started.
1. For greater beauty, let's draw the first line horizontally (Figure 1). Then, we will draw a wide arc with a radius R=a, i.e.
Origami paper pyramid
with a radius equal to the length of the lateral edge of the pyramid. We get point A. From it we make a notch on the arc with a compass, with a radius r \u003d b (the length of the side of the base of the pyramid). Let's get point B. We already have the first face of the pyramid!
2. From point B we make another notch with the same radius - we get point C and connecting it with points B and S we get the second side face of the pyramid (Figure 2).
3. Repeating these steps the required number of times (it all depends on how many faces your pyramid has) we will get such a fan (Figure 3). With the correct construction, you should get all the points of the base, and the extreme ones should be repeated.
4. This is not always required, but still it is necessary: add the base of the pyramid to the development of the side surface. I believe that everyone who has read up to this point can draw a six-eight-pentagon (how to draw a pentagon is described in detail in this lesson). The difficulty lies in the fact that the figure needs to be drawn in the right place and at the right angle. Draw an axis through the middle of any face. From the point of intersection with the line of the base, we plot the distance m, as shown in Figure 4.
Drawing a perpendicular through this point, we get the axes of the future hexagon. From the resulting center we draw a circle, as you did when building a top view. Please note that the circle must pass through two points of the side face (in my case, these are F and A)
5. Figure 5 shows the final unfolded view of the hexagonal prism.
This completes the construction of the pyramid sweep. Build your sweeps, learn to find solutions, be corrosive and never give up. Thanks for stopping by. Don't forget to recommend us to your friends 🙂 All the best!
or write down our phone number and tell your friends about us - someone is probably looking for a way to make drawings
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Guys, thanks for helping out! Even the teacher doesn't explain that! Thank you very much!8
Thank you, you're welcome! I have been waiting for something like this all these years :))) Let everything work out!
thank you very much
