Regular quadrangular prism. Everything you need to know about the prism to successfully pass the exam in mathematics (2020) How to make a straight pentagonal prism

A prism is a three-dimensional figure, a polyhedron, of which there are many types: positive and irregular, straight and oblique. According to the figure lying at the base, the prism is from triangular to polygonal. It's easier for everyone to make a straight prism, but above the inclined one you need to work a little harder.

You will need

  • - compasses;
  • - ruler;
  • - pencil;
  • - scissors;
  • - glue;
  • - paper or cardboard.

Instructions

1. Draw the bases of the prism, in this case they will be 2 hexagons. In order to draw the correct hexagon, use a compass. Draw a circle for them, and with the help of the same radius, divide the circle into six parts (for a true hexagon, the sides are equal to the radius of the circumscribed circle). The resulting figure resembles a honeycomb cell. Draw the wrong hexagon randomly, but with the help of a ruler.

2. Now start designing the pattern. The walls of the prism are parallelograms and you need to draw them. In a straight model, the parallelogram is a light rectangle. And its width will always be equal to the side of the hexagon lying at the base of the prism. With a correct figure at the base, all the faces of the prism will be equal to each other. If it is wrong, only one parallelogram (one side face), suitable in size, will correspond to the entire side of the hexagon. At the same time, follow the sequence of the dimensions of the faces.

3. On a horizontal line, place 6 line segments equal to the side of the base of the hexagon in steps. From the points obtained, draw perpendicular lines of the required height. Combine the ends of the perpendiculars with a 2nd horizontal line. You now have 6 rectangles joined together.

4. Attach the 2 hexagons constructed earlier to the bottom and top side of one of the rectangles. To any base if it is positive, and to the corresponding length if the hexagon is incorrect. Circle the silhouette with a solid line, and the fold lines inside the shape with a dotted line. You now have a surface scan of a straight prism.

5. Leave the base the same to create a tilted prism. Draw a parallelogram side that will be one of the faces. There should be six such faces, as you remember. In order now to draw a sweep of an inclined prism, it is necessary to arrange six parallelograms in a further order: three in ascending order, so that their oblique sides form one line, then three in descending order with the same condition. The slope of the resulting line is directly proportional to the tilt of the prism.

6. Draw small trapezoidal overlaps on the short sides to glue the shape, and also on one free long side to the five rectangles in the flat pattern. Cut the blank for the prism along with the overlaps and glue the model.

A prism is a device that separates typical light into separate colors: scarlet, orange, yellow, green, blue, blue, violet. It is a translucent object with a flat surface that refracts light waves, depending on their lengths, and therefore allows you to see light in different colors. Make prism by yourself is easy enough.

You will need

  • Two sheets of paper
  • Foil
  • Glass
  • Compact disc
  • Coffee table
  • Lantern
  • Pin

Instructions

1. A prism can be made from a simple glass. Fill a glass with water a little larger than half. Place the glass on the edge of the coffee table so that about half of the bottom of the glass hangs in the air. At the same time, make sure that the glass is stable on the table.

2. Place two sheets of paper, one by one, next to the coffee table. Turn on the flashlight and shine the rays of light through the glass, so that it falls on the paper.

3. Adjust the position of the lantern and paper until you see a rainbow on the sheets - this is how your ray of light is decomposed into spectra.

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The basic skill of an artist in academic drawing is the knowledge to depict on a plane the simplest volumetric geometric shapes - a cube, prism , cylinder, cone, pyramid and ball. Possessing this skill, it is allowed to build more difficult, combined volumetric forms of architectural and other objects. A prism is a polyhedron, two faces (bases) of which have the same shape and are parallel to each other. The side faces of the prism are parallelograms. According to the number of side faces, the prisms can be 3-, tetrahedral, etc.

You will need

  • - drawing paper;
  • - primitive pencils;
  • - easel;
  • - a prism or an object in the form of a prism (a wooden block, box, casket, part of a children's designer, etc.), preferably white.

Instructions

1. Erect prism it is permissible by inscribing it either in a parallelepiped or in a cylinder. The core difficulty in drawing a prism is the positive construction of the shape of 2 faces of its base. When drawing a prism lying on one of the side faces, there is an additional difficulty in observing the laws of perspective, because in such an arrangement the perspective reduction of the side faces becomes noticeable.

2. When drawing a vertical prism, start by marking its central axis - a vertical line drawn in the middle of the sheet. On the axis line, sweep the center of the top (visible) face of the base and draw a horizontal line through this point. Determine the ratio of the height and width of the prism by the sighting method: look at nature, covering one eye, and holding a pencil in an outstretched hand on the tier of the eyes, sweep the width of the prism visible from your point of view with your finger on a pencil and mentally set this distance along the line of the height of the prism a certain number times (how many times).

3. When measuring the segments with a pencil closer in the drawing, mark the width and height of the prism with dots on the 2 lines drawn earlier, observing the ratio obtained. Draw an ellipse around the center of the top face. Be diligent to accurately convey its imaginary form, looking at nature. Draw approximately the same ellipse (but less flattened) in the plane of the lower face of the prism base. Combine the resulting ellipses with two vertical lines.

4. Now on the upper ellipse it is necessary to mark the segments of the intersection of the side faces and its bases. Looking at nature, sweep the points - the vertices of the polygon - lying at the base of the prism, as you see them, and join them step by step. From these points, draw lines down to the intersection with the lower ellipse. Combine the resulting intersection points as well. During the subsequent drawing, the faces, visible from the selected point of view, are erased or shaded, therefore, draw all auxiliary construction lines without pressing.

5. Lying on its side prism draw with the help of the auxiliary box. Focusing on nature, draw a parallelepiped, observing the theses of perspective - the lines of the lateral edges, when mentally extended to the horizon line, which is invariably located on the tier of the viewer's eyes, converge at one point. Consequently, the (noticeable) edge that is far from us will be slightly smaller than the front one. When determining the aspect ratio of the parallelepiped, use the arm's length (or sighting) method.

6. On the front and back square edges, sweep the vertices of the polygons at the base of the prism and draw them. Combine these points in pairs on 2 faces - draw the side edges of the prism. Remove obscene lines. Highlight the lines of edges and corners of the prism closer to you more thickly, and mark the distant ones with light lines.

7. Looking at the nature, determine the angle of incidence of the light, the clearest, most shaded edges and, with the help of shading of different intensities, convey these light ratios in the drawing. Draw a drop shadow from the subject. Underline the contact line between the prism and the table with the darkest line. Please note that the light reflected from the table surface (reflex) falls on the most shaded edge of the prism from below, and illuminates it slightly. When applying hatching to this face, take into account given result and apply a less saturated tone in the place of the reflex.

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A prism is a polyhedron formed by any final number of faces, two of which - the bases - must necessarily be parallel. Any straight line drawn perpendicular to the bases contains a segment connecting them, called the height of the prism. If all side faces are adjacent to both bases at an angle of 90 °, the prism is called straight .

You will need

  • Prism drawing, pencil, ruler.

Instructions

1. IN straight prism any lateral rib is perpendicular to the base by definition. And the distance between the parallel planes of the side faces is identical at every point, including those points where the side edge is adjacent to them. From these 2 circumstances it follows that the length of the edge of any side face straight prism is equal to the height of this volumetric figure. This means that if you have a drawing that shows such a polyhedron, there are more segments (edges of the side faces) on it, all of which can be designated as the height of the prism. If it is not prohibited by the conditions of the task, primitively designate any side edge as a height, and the task will be solved.

2. If you need to draw a height that does not coincide with the side edges in the drawing, draw a line segment parallel to any of these edges that connects the bases. It is not invariably allowed to do this "by eye", therefore, build two auxiliary diagonals on the side faces - combine a pair of any corners on the upper and the corresponding pair on the lower base. After that, measure any comfortable distance on the upper diagonal and put a point - this will be the intersection of the height with the upper base. Measure the same distance on the lower diagonal and put a second point - the intersection of the height with the lower base. Join these points with a line, and building the height straight prism will be completed.

3. The prism can be depicted taking into account perspective, that is, the lengths of the identical edges of the figure can have different lengths in the figure, the side faces can adjoin the bases at different and not strictly right angles, etc. In this case, in order to correctly observe the proportions, proceed in the same way as described in the previous step, but put the points on the upper and lower diagonals correctly in their middle.

In detail - how to fold a sheet of paper and cut out a beautiful snowflake.

You will need

  • A sheet of paper, I have an ordinary A4 sheet, it's better to take huge napkins
  • Scissors

Instructions

1. We fold the sheet across in half

2. Now twice, just to find the middle

3. We wrap the edges of the paper folded in half, one by one - as seen in the photo

4. We make sure that the leaf is bent evenly, and the ends reach the folds.

5. Now we fold the resulting envelope in half. It is necessary to practice in order to ensure that the outer edge of the sheet reaches exactly to the fold.

6. While there is no skill, it is cooler to draw an approximate silhouette of a snowflake in advance.

7. Cut it out neatly along the silhouette.

8. Expanding diligently.

Note!
Remember that it is impossible to make a through cut, the snowflake will fall apart.

Helpful advice
The thinner the paper, the easier it is to cut the snowflake. It is allowed to make snowflakes from foil.

Note!
In the sweep of an inclined prism, do not draw its edges at too great an angle; on the contrary, the model will be unstable.

Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles

Side rib is the common side of two adjacent side faces

Prism height is a segment perpendicular to the bases of the prism

Diagonal prism - a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane - a plane that passes through the diagonal of the prism and its lateral edges

Diagonal section - the boundaries of the intersection of the prism and the diagonal plane. The diagonal section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section) is the intersection of a prism and a plane drawn perpendicular to its lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

  • Bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
  • Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
  • Lateral surface - the sum of the areas of all lateral faces of the prism
  • Full surface - the sum of the areas of all bases and side faces (the sum of the area of \u200b\u200bthe side surface and bases)
  • Side ribs AA 1, BB 1, CC 1 and DD 1.
  • Diagonal B 1 D
  • Base diagonal BD
  • Diagonal section BB 1 D 1 D
  • Perpendicular section A 2 B 2 C 2 D 2.

Properties of a regular quadrangular prism

  • The bases are two equal squares
  • The bases are parallel to each other
  • The side faces are rectangles
  • The side faces are equal to each other
  • Side faces are perpendicular to the bases
  • The side ribs are parallel and equal
  • Perpendicular section perpendicular to all side edges and parallel to the bases
  • Perpendicular section corners - straight
  • The diagonal section of a regular quadrangular prism is a rectangle
  • Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " correct quadrangular prism "it is understood that:

Correct prism - a prism at the base of which a regular polygon lies, and the side edges are perpendicular to the base planes. That is, a regular quadrangular prism contains at its base square... (see above properties of a regular quadrangular prism) Note... This is part of the lesson with geometry problems (section stereometry - prism). Here are the tasks that cause difficulties in solving. If you need to solve a problem in geometry, which is not here - write about it in the forum. To denote the action of square root extraction in problem solutions, the symbol√ .

A task.

In a regular quadrangular prism, the base area is 144 cm 2, and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Decision.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal to

√144 \u003d 12 cm.
Whence the diagonal of the base of a regular rectangular prism will be
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms a right-angled triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√ ((12√2) 2 + 14 2) \u003d 22 cm

Answer: 22 cm

A task

Determine the full surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of the side face is 4 cm.

Decision.
Since there is a square at the base of a regular quadrangular prism, we will find the side of the base (denoted as a) by the Pythagorean theorem:

A 2 + a 2 \u003d 5 2
2a 2 \u003d 25
a \u003d √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 \u003d 4 2
h 2 + 12.5 \u003d 16
h 2 \u003d 3.5
h \u003d √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S \u003d 2a 2 + 4ah
S \u003d 25 + 4√12.5 * √3.5
S \u003d 25 + 4√43.75
S \u003d 25 + 4√ (175/4)
S \u003d 25 + 4√ (7 * 25/4)
S \u003d 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

The geometric body - prisms are based on polygons, and each side face is a parallelogram. The uninitiated may have been a little scared. But if your child is asked to come to class with a prism, you will naturally want to help him and explain how to make a prism out of paper.

Let's start by making a straight prism. In this prism, the lateral ribs are perpendicular to the bases. The easiest to make with your own hands is a paper prism with three faces, since its bases are the simplest of polygons - triangles. Let's make the "correct" prism. Its bases are represented by equilateral triangles.

Triangular prism

Let's think about the height of our triangular prism made of paper. Let's draw a rectangle - with one side equal to the height, and the other equal to the length of the perimeter of the triangle at the base. The resulting rectangle is divided by parallel lines into three equal parts. From the corners of the rectangle located in the middle, draw circles with a compass with a radius equal to the side of our triangle at the base. Where the circles intersect outside the original rectangle, place points and connect them to the centers of the circles. We should get the shape shown in the middle of the picture. Next, cut out the figure with small allowances for gluing, bend it along the existing straight lines and get the finished prism.

What template is used to make a prism from paper with four faces is clearly demonstrated by the diagram in the figure.

Hexagonal prism

An example of a blank for a pentahedral prism is shown in the figure. Here the height of the pyramid is 10 cm, the length of the sides at the base of the pentahedron is 3 cm. In a similar way, a hexagonal prism can be made of paper, but at its base there is a hexagon.

Oblique prism

An oblique paper prism is shown in this figure. Its side faces are at an angle to the base. Such a prism can be made using a scan pattern.

This image is a "regular" street photo. Overpasses lead the eye to the image ... through the prism

A key element of any photography is how you use light. This article will show you how to split it. The use of a prism when photographing provides new possibilities and is another way of using light refraction.

What does a prism do with light?

Since a prism is a glass object, light refracts as it passes through it, creating several effects that you can use in photography.

There are two ways to use a prism.

  • Rainbow projection - a prism, and in particular its triangular shape, acts by separating light and revealing waves of different lengths in the form of a rainbow. And already you can photograph it.
  • Light redirection - Light can change direction abruptly as it travels through a prism. This means that when you look through it, you can see the picture at a 90-degree angle to you. This factor makes it possible to create a double exposure.

The image clearly shows the rainbow light from the prism, as well as the remnants of light emitted from different angles.

Using a crystal prism to create a rainbow

A great way to use a prism is to create a rainbow. The larger the prism, the larger the rainbow is. Another way to increase its size is to increase the distance between the prism and the surface on which you are projecting the rainbow. The difference between these options is that with an increase in the aforementioned distance, the rainbow light becomes more diffused and less intense.


Use a prism to create your own rainbow

Also notice how high the sun is in the sky. The angle of incidence of sunlight on the prism affects the angle of the projected rainbow. It's easier to project a rainbow onto the ground at noon. To project the rainbow more horizontally, you need to photograph when the sun is lower in the sky, that is, after sunrise or before sunset.

Rainbow as photo detail

Rainbow light is very colorful and can create an interesting effect when projected onto a surface. Look for a surface that is neutral in color (such as gray or white). Pay attention to surfaces with pleasant texture.

Twist the prism until you can see the rainbow being projected onto the surface you are photographing. You can, of course, take a picture while holding the prism and camera. But it's good if you have a friend to help. Since this is a detailed photo, it is best to use a macro lens, but you can find equally interesting compositions using other lenses.

Rainbow in portrait photography

Undoubtedly one of the most popular forms of prism photography is projecting a rainbow onto a model's face. The rainbow won't end up being big, and it would be nice again if the other person held the prism while you were photographing.

Three images in one frame

You can shoot through the glass those objects that appear inside the prism. Raise the prism and rotate it. You will see images inside. However, they will not be the same that are right in front of you. Depending on how you rotate the glass prism, one or two images will appear. It is with them that you can work to create a single press of the shutter button.

Lens selection

For prism photography, wide angle and macro lenses.

  • The wide-angle lens lets you add a background image to your photo. However, the edge of the prism becomes more visible in the frame. It is not easy to blur an image with the aperture available on most wide-angle lenses.
  • Macro lens. Most of prism photography is done using it, as this lens allows you to focus close to the prism and avoid getting your hand caught in the frame. The transition from the background to the image in the prism is also more difficult to detect.

The image was taken with a macro lens with a prism, and in the end it looks like an optical illusion

Aperture for prism photography

Which one you use for such photographs depends mainly on what you plan to do with the background and how sharp you want the image in the prism.

An open aperture of f / 2.8 or faster will certainly work to blur the background. For most photographs, to achieve a multiple exposure feel. This means that an aperture of around f / 8 is the right balance between background and detail, and avoids too sharp a prism line when going to the background.

Background image

Due to the small width of the prism, even with a macro lens, the background takes up most of the frame. So what works as a background for this type of photography?

  • Leading lines - the background that draws attention to the images inside the prism is used effectively. It can be a tunnel or a road that goes to infinity.
  • Texture Background - More of a blank canvas for images in the prism. It could be a brick wall or leaves and flowers.
  • Symmetry. Since a prism divides your image down the middle, using symmetry on both sides of that division is a fairly effective strategy.

Using background symmetry can work well in prism photography

Image in glass

Now the hardest part is getting a good image inside the prism. The images in it can be at 90 degrees to the way you look, or perhaps at 60 degrees to the edge and front of where the photographer is standing. Incorporating this into background composition is a tricky aspect of prism photography.

  • Composition - You already have a good composition for your background. Now you need to save it while adding a point of interest that looks good through the lens. Just use trial and error. Change the angle of inclination of the prism or rotate it; you can also try walking back and forth.
  • Adding a model. An easier way to add interest to a prism image is to make it a portrait photograph. The advantage is that you can simply ask the model to stand in the desired position, from which the refracted light passes through the prism.

Adding a model to the composition of this image made the cherry blossom photo much more interesting.

Use fractals

Fractals are another element that uses refraction in photography. They produce prismatic effects, but are not triangular by themselves. You can take pictures through them without worrying about images being 90 degrees to you. Fractals are often used to create creative portrait photos with soft edges or other abstract images.

Time to go and share the light!

If you want to try something new in photography, you will definitely love it. It's a little tricky to take pictures with, but that's what makes it really fun. Now is the time to take the crystal prism in hand and go to meet the experiments!

Given:
Intersection of pyramid and prism
It is necessary:
Construct a scan of a straight prism and show on it the line of intersection of the prism with the pyramid.

Constructing a straight prism sweep is much easier than a pyramid sweep.

Building a prism sweep

The construction of a sweep of a straight prism is facilitated by the fact that all dimensions for sweeping are taken from the diagrams and we do not need to find the natural values \u200b\u200bof the edges of the prism. Since a straight prism is given, the side edges of the prism are projected onto the frontal projection plane in full size. The base edges of the straight prism are parallel to the horizontal projection plane and are also projected onto it in full size.

Algorithm for constructing a prism sweep

  • We draw a horizontal line.
  • From an arbitrary point G of this straight line, we lay off the segments GU, UE, EK, KG equal to the lengths of the sides of the base of the prism.
  • Perpendiculars are restored from points G, U, ... and values \u200b\u200bequal to the height of the prism are laid on them. The resulting points are connected by a straight line. Rectangle GG1G1G is a flat surface of the prism. To indicate on the sweep of the prism faces, perpendiculars are restored from the points U, E, K.
  • To obtain a full sweep of the prism surface, polygons of its bases are attached to the surface sweep.

To build on the sweep the line of intersection of the prism with the pyramid of closed broken lines 1, 2, 3 and 4, 5, 6, 7, 8, use vertical straight lines.

In more detail in the video tutorial on descriptive geometry in AutoCAD