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African applied arts. African painting

In this lesson you will learn how to draw regular 3D pyramids with a pencil step by step. I've always dreamed of going to Egypt and climbing the ancient pyramid. And you?

We will draw a pyramid using the following concepts: overlap, horizon line, shadows and shading. This tutorial will also help you practice applying smooth, one-sided eyeshadow. Because the sides of a pyramid are flat, they require consistent shading, unlike cylinders, flags, and other curved surfaces, which require progressive shading from dark to light. Let's get started.

1. Draw a straight vertical line.

2. Tilt the sides of the pyramid down, keeping the angle between them the same and making the middle line a little longer.

3. Imagining a compass of directions, draw the bottom of the pyramid in the NW and NE directions.

4. Anchor the pyramid to the sand using the horizon line. Position the light source and draw lines in the SW direction for the cast shadow.

5. Now add a solid, smooth shading to one side of the pyramid opposite the light source.

6. You can stop here, you have a great pyramid! Or you can add a brick texture, with crumbling edges and piled up stone fragments, and you get ruins. I'm thinking about adding doors. Strange? Stupid? Sketch the position of the doors.

7. On the door on the right there is a slope on the right, on the door on the left there is a slope on the left. Draw the slopes on the right side.

8. Now on the left.

9. Complete the shading on the sides opposite the light source. Remember, this is a flat surface which means smooth, solid shading without shading. However, the inside of the curved door on the right side needs to be shaded because on curved surfaces you should always shade the shadow from dark to light, and on flat surfaces you should add solid shading on the sides facing away from the light source.

LESSON 19: PRACTICAL TASK

It all depends on how much time you can devote to this wonderful scene with many pyramids. Notice how one pyramid is below the horizon and several pyramids at a distance, beyond the horizon. A very interesting note in the drawing laws is that overlap always wins over the other eight laws. Look, size doesn't matter in this picture. Usually, objects that we draw larger appear closer, and objects that are smaller appear further away. However, in this picture, even though the huge pyramid is overshadowed by the small ones, it still looks further, deeper in the picture. Why? Due to the force of overlap. Small pyramids are drawn in front of the big one, thus creating the illusion that the giant one is further away.

One of greatest miracles world - the pyramids of Egypt, have survived to this day. Moreover, every year many tourists go to Giza just to see the pyramids and the Sphinx, which seems to be guarding the buildings, frozen motionless among the ruins of the ancient temple.

To interest your child, you can tell him the following:

  • Until now, no one knows how and why the pyramids were built. Scientists have found that this wonder of the world was built much earlier than the kingdom of Egypt appeared.
  • Despite the fact that in ancient times there was no modern technology, the pyramids were built with mathematical precision.
  • For three thousand years, the pyramid of Pharaoh Cheops was the tallest building in the world.

Drawing pyramids is very simple. Follow our instructions and you will succeed. You will need: a sheet of paper; pencil; eraser; ruler;
Step 1

Place for pyramids

First, draw a rectangle in which we will place the pyramids. To make the process easier, you can use a ruler.
Next, divide the rectangle into three parts. You should end up with two rectangles on the left and right, and a square in the center, slightly larger than the rectangles.

Step 2

First pyramid

At this stage we begin to draw the pyramids. First, we draw a pyramid in the first rectangle. It is also called the Pyramid of Khafre. It is the second largest and is in front of the rest. Notice that part of the triangle extends beyond the rectangle.


Step 3

Second pyramid

It was the turn of the Great Pyramid or the Cheops Pyramid. It starts in the first rectangle and extends beyond the edges of the second. The top of the triangle touches the top side of the square.

Step 4

Third pyramid

The last pyramid of Pharaoh Mikerin is the smallest. We draw it only in a rectangle, without going beyond the edges. Part of this structure is hidden behind the Great Pyramid.

Step 5

Removing extra lines

Now you need to erase all unnecessary lines. Only the pyramids remain. Only the first one is fully drawn, since the rest are hidden behind each other.

Step 6

Drawing bricks

From the tops of the pyramids we draw straight lines down, indicating the corners.
Throughout the entire first pyramid we carry out horizontal lines.
We separate these lines at small intervals vertical lines, indicating bricks.
Draw horizontal lines on the remaining pyramids.
Similarly, we divide the lines to make bricks.

    In general, a pyramid can be drawn quite simply. To begin with, the sheet must be divided in half; this can be done in a line using a ruler. Next we draw the side parts of the pyramid and everything is ready.

    Everything is easy and simple to draw!!

    At school, we all drew pyramids in class, only these were geometric shapes. And in order to draw a pyramid, you will definitely need a ruler and a pencil, and have an eraser to erase what is unnecessary.

    The pyramid should look like this:

    And in order to draw it, we must first draw a square, then draw the edges inside the square and connect them at the base. The lines at the base of the pyramid are highlighted in green for their subsequent connection.

    Then you can paint your pyramid in any color. Either paints or colored markers are suitable for this. And your pyramid will be prettier in color.

    Here is a video master class on how to draw the Pyramid of Cheops along with the Sphinx.

    And below is step by step drawing pyramids, if you wish, you can draw sand and set off a camel caravan and the mysterious Valley of the Pharaohs is ready :)

  • Drawing a pyramid step by step

    Of course, I’m not an artist and it’s difficult to explain to me how to draw a pyramid correctly, but I can show you step by step how masters draw.

    I think that drawing a pyramid is not difficult, looking at these sketches.

    • Draw a picture of a pyramid. We will do this in seven steps.

    First step. Let's draw a square, the length of its side will be five centimeters.

    Second step. Take a compass and set its length equal to the side square. Now we place the compass needle at the far right point and draw a line from the opposite edge to the top side, as shown in the figure:

    Third step. We do a similar operation from the opposite edge, so we get two curves that intersect at one point.

    Fourth step. From both extreme points base we draw two straight lines intersecting at the intersection point of the curve.

    Fifth step. Erase the curved lines and the sides and tops of the square.

    Sixth step. Draw the side of the pyramid:

    Drawing a pyramid with a pencil is probably one of the easiest drawing tasks. There is no need for any special talent or possession of kakini or drawing skills. I provide an instructive video.

    You can draw a pyramid with a pencil step by step, based on the following diagrams:

    Stages of drawing pyramids:

    1) We start making sketches;

    2) We begin to draw the elements as in the pictures;

    3) Let's move on to detailing the image;

    4) Shade and trace along the contour.

    Pyramids are monuments of Egyptian culture and world culture in general.

    IN ancient Egypt pyramids were the place where pharaohs were buried. And many of them (pyramids) bore/bear the name of the pharaohs who were buried in it. (Cheops, Tutankhamun).

    Now, it is for them that many people go to Egypt. To look at the magnificent buildings of ancient times.

    That's how draw pyramids with a pencil step by step:

    Step one:

    Step two:

    Step three:

    Step four:

    To depict a pyramid, you need to draw two triangles adjacent to each other, for example, as in the video below. And if necessary egyptian pyramid, then you need to add a pattern like brickwork on a pyramid and complete the entrance.

It would seem that what could be difficult or incorrect in the image of a pyramid? Is it really possible that a math tutor can’t do without special techniques and techniques here too? Only 4 points are marked (any 3 of which do not lie on the same straight line) and connected by six segments. That's all. What is there to discuss here? But even in such a simple situation, the math tutor has to correct the student’s mistakes. Not even so much mathematical as strategic. Which? A drawing in which it is impossible to examine or show the elements of a spatial body, to sign the values ​​of quantities, in which it is not possible to expand with additional constructions, is better to redo it. Any tutor should understand this and, at the beginning of the Unified State Exam preparation course, spend some time teaching the rules and culture of drawing. In addition to the requirements for its accuracy and convenient arrangement of information from the task conditions, there are also mathematical laws for its implementation. Let's take a closer look at them.

The rule of the image method.

The image method is a separate subject, the study of which at the Faculty of Mathematics of Moscow State University is devoted to a whole semester. What we draw on paper are traces of projections of body parts onto a certain plane. It determines which segments and which sections will be clearly visible, and which ones will be “creeping” on each other or hidden. When a math tutor decides which side to draw a pyramid for a student, he determines the location of the plane and the direction of projection.

There are geometric laws for the projection of the simplest stereometric objects. The lengths of non-parallel segments, for example, when depicted, can change the ratio of their lengths (it is better for the teacher to say “distorted”). If in reality one of them is larger than the other, then in the projection everything can be exactly the opposite. The same goes for corners. A right angle can be projected into either an acute or an obtuse angle. In order for a math tutor to convince a student of this, he should twirl an ordinary square in front of his eyes. However, the ratio of the lengths of segments lying on parallel or coinciding lines does not change and, in particular, the midpoints of the sides of polygons (pyramid faces) are not distorted. This explains the law of location of the base of the height of a regular triangular pyramid: it must be the point of intersection of its medians(center of gravity). Parallelism is also not distorted. If there is parallelism between straight lines in space, then it is also preserved between their traces. Therefore, a parallelogram is chosen to represent the base of a regular quadrangular pyramid.

Readability of the drawing.

It is important to arrange the pyramid so that all its parts are not just visible, but allow for further complication of the drawing: drawing apothems, traces of sections, etc.

To do this, build, for example, correct pyramid preferably down up through height (this is how it is used in almost all tasks). First, the math tutor draws the base of the pyramid, then its center, and from this point restores the perpendicular. Its top end is selected so that all inclined ribs are sufficiently distant from each other. If you build in the reverse order, you may miss the center of the polygon. Of course, this is not critical for solving problems correctly triangular pyramid, but still unsightly to perceive. Midpoints should be displayed as midpoints.

Construction of the foundation.
Regardless of the type of base of the tetrahedron, it is depicted acute triangle and pull it left or right. For what? If it is isosceles, then one of the side ribs will cover the height (if, of course, its base is correctly located). This is shown in the figure.

Frontal image of a tetrahedron. Tutor's rule.

Which edge is best to depict a pyramid? That is, how to optimally choose a plane for projection? Some teachers and math tutors, unfortunately, do not pay attention to such a “trifle” as the frontal location of the pyramid. But in vain. There are two types of drawing: “with the corner of the base towards us” or “with the corner away from us.” Consider the drawing with the “corner ABC away from us”:
We restore the height from bottom to top and select the position of its end (the top of the pyramid) taking into account the acceptable span of the ABP face. To do this, the most important thing is not to get point P on line AB. Otherwise we won’t see the line. A significant deviation from the point of intersection (in the image) of lines AB and OP causes a rather small deviation of ray AP from ray AB and therefore, in order to achieve the span of edge ABP, it is necessary to choose point P either very low or very high. The latter can overly enlarge the drawing, stretching the pyramid upward (reducing the space for the solution itself), and a low point makes the drawing small. Therefore, I do not recommend that math tutors work on this front. It is best to turn triangle ABC over with the corner facing us.
Note that now the position of point P does not in any way affect the readability of edge ABP and if it is not isosceles and “strongly acute”, but point O is its center of gravity (that is, O is not at the height of the base), then the height of the pyramid will not be covered by edge BP nor at what location of the vertex P. In this case, the mathematics tutor receives a certain freedom in choosing the top of the pyramid, which is extremely important for improving the readability of further constructions in complex problems.

Drawing invisible lines.
A math tutor, of course, can do without the dotted lines. However, what is good for a Russian is death for a German. It is important for the student to perceive the body exactly from the side from which the tutor sees it. Especially when working with edges. I advise the mathematics teacher to more often name the faces not by their vertices, but by their natural location: “near”, “far”, “left”, “right”. If a child forms an image of an object “back to front” in his head, then problems will arise with describing the progress of additional constructions, reading a drawing, and even explaining incomprehensible moments of decisions.

on the construction of a quadrangular pyramid.
The base of a regular quadrangular pyramid should be depicted as a parallelogram. Why? Of course, we can position the square to the projection plane in such a way that the right angles are preserved (and we get a rectangle), but then the apothems of the two closest faces will cover the height of the pyramid. I can’t find any other explanation for the current image standards.

Alexander Kolpakov, mathematics tutor in Moscow. Preparation for the Unified State Exam.

Rectangle, square, triangle, trapezoid and others - geometric shapes from the section exact science. A pyramid is a polyhedron. The base of this figure is a polygon, and the side faces are triangles with a common vertex, or trapezoids. To fully represent and study any geometric object, mock-ups are made. They use a wide variety of materials from which the pyramid is made. The surface of a polyhedral figure, unfolded on a plane, is called its development. The method of converting flat objects into three-dimensional polyhedra and certain knowledge of geometry will help you create a layout. It is not easy to make developments from paper or cardboard. You will need the ability to make drawings to specified dimensions.

Materials and accessories

Modeling and execution of polyhedral volumes geometric shapes- an interesting and exciting process. From paper you can make a large number of all kinds of layouts. To work you will need:

  • paper or cardboard;
  • scissors;
  • pencil;
  • ruler;
  • compass;
  • eraser;
  • glue.

Defining parameters

First of all, let's determine what the pyramid will be like. The development of this figure is the basis for the manufacture volumetric figure. Completing the work will require extreme precision. If the drawing is incorrect, it will be impossible to assemble a geometric figure. Let's say you need to make a layout of the correct

Any geometric body has certain properties. This figure has a base and its vertex is projected into its center. The base is selected. 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 in this case there will be three of them. It is also important to know the sizes of all components, from which the pyramid will be composed. Paper developments are made in accordance with all the data of the geometric figure. The parameters of the future model are agreed upon in advance. The choice of material used depends on these data.

How is a regular pyramid unfolded?

The basis of the model is a sheet of paper or cardboard. Work begins with a drawing of the pyramid. The figure is presented in expanded form. Flat image on paper corresponds to pre-selected sizes and parameters. has a regular polygon as its base and its altitude passes through its center. Let's make it first simple model. In this case, it is a triangular pyramid. Determine the dimensions of the selected figure.

To construct a development of a pyramid, the base of which is a regular triangle, in the center of the sheet, using a ruler and pencil, draw a base of the given dimensions. Next, to each side we draw the side faces of the pyramid - triangles. Now let's move on to their construction. We measure the dimensions of the sides of the triangles on the lateral surface with a compass. We place 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. To glue a three-dimensional figure, valves are provided on the sides of the side faces. We finish drawing small trapezoids.

Layout assembly

Using scissors, cut out the completed design along the contour. Carefully bend the development along all the lines. We tuck the trapezoid valves inside the figure so that its edges close together. We lubricate them with glue. After thirty minutes the glue will dry. The volumetric figure is ready.

First, let's imagine what the geometric figure looks like, the model of which we will make. The base of the selected pyramid is a quadrangle. The side ribs are triangles. For work we use the same materials and devices as in the previous version. We draw the drawing on paper with a pencil. In the center of the sheet we draw a quadrilateral with the selected parameters.

We divide each side of the base in half. We draw a perpendicular, which will be the height of the triangular face. Using a compass solution equal to the length of the side face of the pyramid, we make notches on the perpendiculars, placing its leg at the top of the base. We connect both corners of one side of the base to the resulting point on the perpendicular. As a result, we get a square in the center of the drawing, on the edges of which triangles are drawn. To fix the model on the side faces, add auxiliary valves. For reliable fastening, a strip of centimeter width is enough. The pyramid is ready for assembly.

The final stage of the layout

We cut out the resulting pattern of the figure along the contour. We bend the paper along the drawn lines. The three-dimensional figure is assembled by gluing. Lubricate the provided valves with glue and fix the resulting model.

Volumetric layouts of complex figures

After completing a simple polyhedron model, you can move on to more complex geometric shapes. Unfolding 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, compass and ruler.

Construction of a drawing

The development of a truncated pyramid is performed in several stages. The side face of a truncated pyramid is a trapezoid, and the bases are similar polyhedra. Let's say these are squares. On a sheet of paper we draw a trapezoid with the given dimensions. We extend the sides of the resulting figure until they intersect. The result is an isosceles triangle. We measure its side with a compass. On a separate sheet of paper we build which will be the measured distance.

The next stage is the construction of the side ribs that the truncated pyramid has. The sweep is performed inside the drawn circle. Using a compass, measure the lower base of the trapezoid. On the circle we mark five points that connect the lines to its center. We get four isosceles triangle. Using a compass, measure the side of the trapezoid drawn on a separate sheet. We put this distance on each side of the drawn triangles. We connect the resulting points. The side faces of the trapezoid are ready. All that remains is to draw the upper and lower bases of the pyramid. In this case, these are similar polyhedra - squares. We add squares to the upper and lower bases of the first trapezoid. The drawing shows all the parts that the pyramid has. The scan is almost ready. All that remains is to finish drawing the connecting valves on the sides of the smaller square and one of the faces of the trapezoids.

Completing the Simulation

Before gluing the three-dimensional figure, the drawing along the contour is cut out with scissors. Next, the development is carefully bent along the drawn lines. We tuck the mounting valves inside the model. We lubricate them with glue and press them to the edges of the pyramid. Let the models dry.

Making different models of polyhedra

Making three-dimensional models of geometric shapes - exciting activity. To master it thoroughly, you should start by performing the simplest sweeps. Gradually moving from simple crafts to more complex models, you can begin to create the most intricate designs.



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