How to make a paper prism Volume and surface area of a regular quadrangular prism Formulas for a regular quadrangular prism.

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At the heart of a geometric body - a prism - are 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 a lesson with a prism, you will naturally want to help him and explain how to make a paper prism.

Let's start by making a straight prism. In this prism, the side edges 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 a "correct" prism. Its bases are represented by equilateral triangles.

triangular prism

Let's think about the height of our triangular paper prism. 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, we 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, put points and connect them to the centers of the circles. We should get the figure shown in the middle of the picture. Next, we cut out the figure with small allowances for gluing, bend along the existing straight lines and get the finished prism.

According to what template a paper prism with four faces is made, the diagram in the figure clearly demonstrates.

Hexagonal prism

An example of a blank for a five-sided prism is shown in the figure. Here the height of the pyramid is 10 cm, the length of the sides of the pentahedron at the base is 3 cm. Similarly, a hexagonal paper prism can be made, but at its base lies a hexagon.

tilted prism

An inclined paper prism is shown in this figure. Its side faces are at an angle to the base. Such a prism can be made according to a scanning template.

In the school curriculum for the course of solid geometry, the study of three-dimensional figures usually begins with a simple geometric body - a prism polyhedron. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrangles, to which the sides are perpendicular, having the shape of parallelograms (or rectangles if the prism is not inclined).

What does a prism look like

A regular quadrangular prism is a hexagon, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.

The figure, which depicts a quadrangular prism, is shown below.

You can also see in the picture the most important elements that make up a geometric body. They are commonly referred to as:

Sometimes in problems in geometry you can find the concept of a section. The definition will sound like this: a section is all points of a volumetric body that belong to the cutting plane. The section is perpendicular (crosses the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be built is 2), passing through 2 edges and the diagonals of the base.

If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.

Various ratios and formulas are used to find the reduced prismatic elements. Some of them are known from the course of planimetry (for example, to find the area of the base of a prism, it is enough to recall the formula for the area of a square).

Surface area and volume

To determine the volume of a prism using the formula, you need to know the area of \u200b\u200bits base and height:

V = Sprim h

Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in a more detailed form:

V = a² h

If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:

To understand how to find the lateral surface area of a prism, you need to imagine its sweep.

It can be seen from the drawing that the side surface is made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:

Sside = Pos h

Since the perimeter of a square is P = 4a, the formula takes the form:

Sside = 4a h

For cube:

Sside = 4a²

To calculate the total surface area of a prism, add 2 base areas to the side area:

Sfull = Sside + 2Sbase

As applied to a quadrangular regular prism, the formula has the form:

Sfull = 4a h + 2a²

For the surface area of a cube:

Sfull = 6a²

Knowing the volume or surface area, you can calculate the individual elements of a geometric body.

Finding prism elements

Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, formulas can be derived:

base side length: a = Sside / 4h = √(V / h);
height or side rib length: h = Sside / 4a = V / a²;
base area: Sprim = V / h;
side face area: Side gr = Sside / 4.

To determine how much area a diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:

Sdiag = ah√2

To calculate the diagonal of the prism, the formula is used:

dprize = √(2a² + h²)

To understand how to apply the above ratios, you can practice and solve a few simple tasks.

Examples of problems with solutions

Here are some of the tasks that appear in the state final exams in mathematics.

Exercise 1.

Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it into a container of the same shape, but with a base length 2 times longer?

It should be argued as follows. The amount of sand in the first and second containers did not change, i.e., its volume in them is the same. You can define the length of the base as a. In this case, for the first box, the volume of the substance will be:

V₁ = ha² = 10a²

For the second box, the length of the base is 2a, but the height of the sand level is unknown:

V₂ = h(2a)² = 4ha²

Because the V₁ = V₂, the expressions can be equated:

10a² = 4ha²

After reducing both sides of the equation by a², we get:

As a result, the new sand level will be h = 10 / 4 = 2.5 cm.

Task 2.

ABCDA₁B₁C₁D₁ is a regular prism. It is known that BD = AB₁ = 6√2. Find the total surface area of the body.

To make it easier to understand which elements are known, you can draw a figure.

Since we are talking about a regular prism, we can conclude that the base is a square with a diagonal of 6√2. The diagonal of the side face has the same value, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.

The length of any edge is determined through the known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found by the formula for the cube:

Sfull = 6a² = 6 6² = 216

Task 3.

The room is being renovated. It is known that its floor has the shape of a square with an area of 9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?

Since the floor and ceiling are squares, that is, regular quadrilaterals, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of its lateral surface.

The length of the room is a = √9 = 3 m.

The square will be covered with wallpaper Sside = 4 3 2.5 = 30 m².

The lowest cost of wallpaper for this room will be 50 30 = 1500 rubles.

Thus, to solve problems on a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as to know the formulas for finding the volume and surface area.

How to find the area of a cube

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

A key element of any photography is how you use light. In this article, you will learn how to split it. The use of a prism in photography provides new opportunities and is another way to use the refraction of light.

What does a prism do to light?

Since the 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 dividing the light and revealing waves of various lengths in the form of a rainbow. And you can take a picture of it.
Light Redirection - Light can change direction abruptly when passing through a prism. This means that when you look through it, you will be able to see the painting at a 90 degree angle to yourself. This factor makes it possible to create a double exposure.

The picture clearly shows the rainbow light from the prism, as well as the remnants of light emitted at 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 resulting rainbow. Another way to increase its size is to increase the distance between the prism and the surface you are projecting the rainbow onto. The difference between these options is that as the aforementioned distance increases, the rainbow light becomes more diffuse and less intense.

Using a prism, you can create your own rainbow

Notice also how high the sun is in the sky. The angle at which sunlight strikes the prism affects the angle of the projected rainbow. It is easier to project a rainbow onto the ground at noon. To project a rainbow more horizontally, you need to photograph when the sun is lower in the sky, i.e. after sunrise or before sunset.

rainbow as photo detail

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

Spin the prism until you can see the rainbow projected onto the surface you are photographing. You can, of course, take a picture by holding the prism and the camera. But it's good if you have a friend to help. Since this is a detailed photo, it is better 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 the model's face. The rainbow won't end up being big, and it would be nice, again, to have another person hold the prism while you take the picture.

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 as those directly in front of you. Depending on how you rotate the glass prism, one or two images will be visible. These are the ones you can work with to create a one-click shutter.

Lens selection

For prism photography - wide angle and macro lenses.

The wide-angle lens allows you to add a background image to your photo. However, the edge of the prism becomes more visible in the frame. It's not easy to blur an image with the aperture available on most wide-angle lenses.
macro lens Most prism photography is done with it, as this lens allows you to focus close to the prism and avoid trapping your hand in the frame. The transition from the background to the prism image is also harder 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 these photos depends mostly on what you plan to do with the background and how sharp you want the image to be in the prism.

An open aperture of f/2.8 or more will certainly work to blur the background. Most photographs to achieve the feeling of multiple exposure. This means that an aperture of around f/8 is the right balance between background and detail, and avoids the prism line being too harsh when transitioning 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 backdrop for this type of photo?

Leading lines - the background that draws attention to the images inside the prism - is used effectively. It could be a tunnel or a road leading to infinity.
The texture background is more of a blank canvas for images in a prism. It can be a brick wall or leaves and flowers.
Symmetry. Since the prism splits your image down the middle, using symmetry on both sides of that split is a pretty effective strategy.

Using background symmetry can work well in prism photography.

Image in glass

Now the hard part is getting a good image inside the prism. The images in it may be at 90 degrees to where you are looking, or perhaps at 60 degrees to the edge and in 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 we need to save it while adding a point of interest that would look good through a prism. Just use trial and error. Change the angle of the prism or rotate it; You can also try stepping back and forth.
Adding a model. An easier way to add interest to an image in a prism is to take it as a portrait photo. 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 sakura photo much more interesting.

Use fractals

Fractals are another element that uses refraction in photography. They produce prismatic effects, but are not triangular in themselves. You can shoot through them without worrying about the images being at a 90 degree angle to you. Fractals are often used to create creative soft-edged portrait photos or other abstract shots.

Time to go and share the light!

If you want to try something new in photography, you will definitely love . It's a bit difficult to photograph with her, but that's what makes the process really interesting. Right now it's time to take a crystal prism in your hands and go towards experiments!

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 line segment perpendicular to the bases of the prism

Prism Diagonal- 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 side 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)- this is the intersection of a prism and a plane drawn perpendicular to its side edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are marked with 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 the side faces of the prism
Total surface - the sum of the areas of all bases and side faces (the sum of the area of the 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 sides are rectangles.
Side faces are equal to each other
Side faces are perpendicular to the bases
Lateral ribs are parallel to each other and equal
Perpendicular section perpendicular to all side ribs and parallel to the bases
Perpendicular Section Angles - Right
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 " regular quadrangular prism" implies that:

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see above the properties of a regular quadrangular prism) Note. This is part of the lesson with tasks in geometry (section solid geometry - 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 extracting a square root in solving problems, the symbol is used√ .

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.

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

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

The diagonal of a regular prism forms a right 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 ) = 22 cm

Answer: 22 cm

A task

Find the total surface area of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of the side face is 4 cm.

Solution.
Since the base of a regular quadrangular prism is a square, then the side of the base (denoted as a) is found by the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √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 = 16
h 2 \u003d 3.5
h = √3.5

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

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

Answer: 25 + 10√7 ≈ 51.46 cm 2.