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 how high ours will be triangular prism from 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, 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 prism of paper 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.

AT school curriculum in 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 quadrilaterals, 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- 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 ( maximum amount sections that can be built - 2) passing through 2 edges and 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 new level sand 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 quadrangles, 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 for 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

Given:
Intersection of pyramid and prism
Necessary:
Build a sweep of a straight prism and show on it the line of intersection of the prism with the pyramid.

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

Construction of a prism scan

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

Algorithm for constructing a prism scan

• We draw a horizontal line.
• From an arbitrary point G of this line, we set aside the segments GU, UE, EK, KG equal to the lengths of the sides of the base of the prism.
• From points G, U, ..., perpendiculars are restored and quantities equal to the height of the prism are laid on them. The resulting points are connected by a straight line. Rectangle GG1G1G is a development of the side surface of the prism. To indicate on the development of the faces of the prism from the points U, E, K, perpendiculars are restored.
• To obtain a complete development of the surface of the prism, the polygons of its bases are attached to the development of the surface.

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

More details in the video tutorial on descriptive geometry in AutoCAD

A prism is a geometric body, a polyhedron, the bases of which are equal polygons, and the side faces are parallelograms. To the uninitiated, this may sound a bit intimidating. And, when your child needs to bring a prism made at home to a geometry lesson, you are at a loss, not knowing how to help your beloved child. In fact, everything is not so difficult and, using our tips on how to make a prism, you will adequately cope with this problem.

How to make a paper prism

We will immediately agree that we will do a straight prism, that is, a prism in which the side edges will be perpendicular to the bases. Do the same inclined prism paper is very problematic (such layouts are usually made of wire).

We already know that two identical polygons lie at the bases of a prism. Therefore, our work will begin with them. The simplest of the polygons is the triangle. This means that we will first make a triangular prism.

How to make a triangular prism

We will need thick white paper for drawing, a pencil, a protractor, compasses, a ruler, scissors and glue.

We draw a triangle, any one is possible, but to make our prism especially beautiful, we will make the triangle equilateral. Such a prism in geometry is called "correct". We choose at our discretion the size of the side of the triangle, let's say 10 cm. With a ruler we put this segment on paper and with a protractor we measure an angle of 60 ∗ from one end of our segment.

We draw an inclined line. On it, using a ruler, set aside 10 cm from the end of the segment. Thus, we have found the third vertex of the triangle. We connect this point with the ends of the initial segment and the equilateral triangle is ready. It can be cut out. Similarly, we make the second triangle, or carefully trace the contours of the first on paper. Well, we already have two reasons.

We make the side edges. We decide what the height of the prism will be. Let's say 20 cm. We draw a rectangle in which the value of one side is the height of the prism (in our case, 20 cm), and the second side is equal to the value of the side of the base multiplied by the number of these sides (we have: 10 cm x 3 = 30 cm) .

On the long sides we make marks every 10 cm. We connect the opposite marks with straight lines. On them then it will be necessary to carefully bend the paper. These are the side edges of our prism. We outline narrow allowances for gluing along two long and one short sides of the rectangle (1 cm wide strips are enough). We cut out the rectangle along with the allowances, carefully bend them according to the markup. We bend the ribs.

We start assembly. We glue the rectangle along the side face into a tube of triangular section. Glue base triangles on top and bottom on the bent allowances. The prism is ready.

It is probably not worth going into the details of the question of how to make a prism out of cardboard. The entire assembly algorithm remains the same, only replace the paper with thin cardboard. By changing the number of sides of the base polygons, you can now independently make both a five- and a hexagonal prism.

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

Diagonal right 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.