The total surface area of the prism is called the sum. Prism base area: from triangular to polygonal
General information about straight prism
The lateral surface of a prism (more precisely, the lateral surface area) is called sum areas of the side faces. The total surface of the prism is equal to the sum of the lateral surface and the areas of the bases.
Theorem 19.1. The lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length lateral rib.
Proof. The lateral faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. It follows that side surface prism is equal
S = a 1 l + a 2 l + ... + a n l = pl,
where a 1 and n are the lengths of the base edges, p is the perimeter of the base of the prism, and I is the length of the side edges. The theorem has been proven.
Practical task
Problem (22) . In an inclined prism it is carried out section, perpendicular to the side ribs and intersecting all the side ribs. Find the lateral surface of the prism if the perimeter of the section is equal to p and the side edges are equal to l.
Solution. The plane of the drawn section divides the prism into two parts (Fig. 411). Let us subject one of them to parallel translation, combining the bases of the prism. In this case, we obtain a straight prism, the base of which is the cross-section of the original prism, and the side edges are equal to l. This prism has the same lateral surface as the original one. Thus, the lateral surface of the original prism is equal to pl.
Summary of the covered topic
Now let’s try to summarize the topic we covered about prisms and remember what properties a prism has.
Prism properties
Firstly, a prism has all its bases as equal polygons;
Secondly, in a prism all its lateral faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all lateral edges are equal;
Also, it should be remembered that polyhedra such as prisms can be straight or inclined.
Which prism is called a straight prism?
If the side edge of a prism is located perpendicular to the plane of its base, then such a prism is called a straight one.
It would not be superfluous to recall that the lateral faces of a straight prism are rectangles.
What type of prism is called oblique?
But if the side edge of a prism is not located perpendicular to the plane of its base, then we can safely say that it is an inclined prism.
Which prism is called correct?
If a regular polygon lies at the base of a straight prism, then such a prism is regular.
Now let us remember the properties that a regular prism has.
Properties of a regular prism
Firstly, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, they are always equal rectangles;
Thirdly, if you compare the sizes of the side ribs, then in a regular prism they are always equal.
Fourthly, a correct prism is always straight;
Fifthly, if in a regular prism the lateral faces have the shape of squares, then such a figure is usually called a semi-regular polygon.
Prism cross section
Now let's look at the cross section of the prism:
Homework
Now let's try to consolidate the topic we've learned by solving problems.
Let's draw an inclined triangular prism, the distance between its edges will be equal to: 3 cm, 4 cm and 5 cm, and the lateral surface of this prism will be equal to 60 cm2. Having these parameters, find the side edge of this prism.
Do you know that geometric figures constantly surround us not only in geometry lessons, but also in Everyday life There are objects that resemble one or another geometric figure.
Everyone at home, at school or at work has a computer, system unit which has the shape of a straight prism.
If you pick up a simple pencil, you will see that the main part of the pencil is a prism.
Walking along the central street of the city, we see that under our feet lies a tile that has the shape of a hexagonal prism.
A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions
“Lesson Pythagorean theorem” - Pythagorean theorem. Determine the type of quadrilateral KMNP. Warm up. Introduction to the theorem. Determine the type of triangle: Lesson plan: Historical excursion. Solving simple problems. And you will find a ladder 125 feet long. Calculate the height CF of the trapezoid ABCD. Proof. Show pictures. Proof of the theorem.
“Prism volume” - The concept of a prism. Straight prism. The volume of the original prism is equal to the product S · h. How to find the volume of a straight prism? The prism can be divided into straight triangular prisms with height h. Drawing the altitude of triangle ABC. The solution of the problem. Lesson objectives. Basic steps in proving the direct prism theorem? Study of the theorem about the volume of a prism.
“Prism polyhedra” - Give the definition of a polyhedron. DABC – tetrahedron, convex polyhedron. Application of prisms. Where are prisms used? ABCDMP is an octahedron made up of eight triangles. ABCDA1B1C1D1 – parallelepiped, convex polyhedron. Convex polyhedron. The concept of a polyhedron. Polyhedron А1А2..АnB1B2..Bn - prism.
“Prism 10th grade” - A prism is a polyhedron whose faces are in parallel planes. Using prisms in everyday life. Sside = Base + h For a straight prism: Sp.p = Pbas. h + 2Sbas. Inclined. Correct. Straight. Prism. Formulas for finding area. Application of prism in architecture. Sp.p = Sside + 2Sbase
"Proof of the Pythagorean Theorem" - Geometric proof. The meaning of the Pythagorean theorem. Pythagorean theorem. Euclid's proof. "IN right triangle The square of the hypotenuse is equal to the sum of the squares of the legs.” Proof of the theorem. The significance of the theorem is that most of the theorems of geometry can be deduced from it or with its help.
Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Right prism
Theorem 2. The area of the lateral surface of the prism
Parallelepiped:
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped
Prism is a polyhedron whose two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than the bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Lateral ribs edges that do not belong to the bases are called. The union of lateral faces is called lateral surface of the prism, and the union of all faces is called the full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. 
Direct prism called a prism whose side ribs are perpendicular to the planes of the bases. 
Correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.
Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P^ - perpendicular section perimeter;
S b - lateral surface area;
V - volume;
S p is the area of the total surface of the prism.
|
V=SH |
*It is assumed that every two successive planes intersect and that the last plane intersects the first
Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same holds for triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to AC) like the line of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC is equal to A "C") as opposite sides parallelogram and that the angles formed by these sides are equal and have the same direction.
Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.
Definition 3
. A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to the prismatic surface - side faces; edges of the prismatic surface - side ribs of the prism. By virtue of the previous theorem, the base of the prism is equal polygons. All lateral faces of the prism - parallelograms; all side ribs are equal to each other.
Obviously, if the base of the prism ABCDE and one of the edges AA" in size and direction are given, then it is possible to construct a prism by drawing edges BB", CC", ... equal and parallel to edge AA".
Definition 4 . The height of a prism is the distance between the planes of its bases (HH").
Definition 5
. A prism is called straight if its bases are perpendicular sections of the prismatic surface. In this case, the height of the prism is, of course, its side rib; the side edges will be rectangles.
Prisms can be classified according to the number of side faces, equal number sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.
Theorem 2
. The area of the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be a given prism and abcde its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. The face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that coincides with ab; the area of the face ВСВ "С" is equal to the product of the base ВВ" by the height bc, etc. Consequently, the side surface (i.e. the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", ВВ", .., for the amount ab+bc+cd+de+ea.
Polyhedra
The main object of study of stereometry is spatial bodies. Body represents a part of space limited by a certain surface.
Polyhedron is a body whose surface consists of a finite number of flat polygons. A polyhedron is called convex if it is located on one side of the plane of every plane polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices are vertices of the polyhedron.
For example, a cube consists of six squares, which are its faces. It contains 12 edges (the sides of the squares) and 8 vertices (the tops of the squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
Prism is a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. Polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are lateral edges of the prism.
Prism height is called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-carbon, if its base contains an n-gon.
Any prism has the following properties, resulting from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The lateral edges of the prism are parallel and equal.
The surface of the prism consists of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of the lateral surface of a prism is the sum of the areas of the lateral faces.
Straight prism
The prism is called straight, if its lateral edges are perpendicular to the bases. Otherwise the prism is called inclined.
The faces of a right prism are rectangles. The height of a straight prism is equal to its side faces.
Full prism surface is called the sum of the lateral surface area and the areas of the bases.
With the right prism called a right prism with a regular polygon at its base.
Theorem 13.1. The area of the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, which is the same, by the lateral edge).
Proof. The lateral faces of a right prism are rectangles, the bases of which are the sides of the polygons at the bases of the prism, and the heights are the lateral edges of the prism. Then, by definition, the lateral surface area is:
,
where is the perimeter of the base of a straight prism.
Parallelepiped
If parallelograms lie at the bases of a prism, then it is called parallelepiped. All faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.
Theorem 13.2. The diagonals of a parallelepiped intersect at one point and are divided in half by the intersection point.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of a parallelepiped are parallelograms, then and , which means according to To there are two straight lines parallel to the third. In addition, this means that straight lines and lie in the same plane (plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals intersect and are divided in half by the intersection point, which was what needed to be proven.
A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. The lengths of the non-parallel edges of a rectangular parallelepiped are called its linear dimensions (dimensions). There are three such sizes (width, height, length).
Theorem 13.3. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three dimensions 
(proven by applying Pythagorean T twice).
A rectangular parallelepiped with all edges equal is called cube.
Tasks
13.1 How many diagonals does it have? n-carbon prism
13.2 In an inclined triangular prism, the distances between the side edges are 37, 13 and 40. Find the distance between the larger side edge and the opposite side edge.
13.3 A plane is drawn through the side of the lower base of a regular triangular prism, intersecting the side faces along segments with an angle between them. Find the angle of inclination of this plane to the base of the prism.
IN school curriculum In a stereometry course, the study of three-dimensional figures usually begins with a simple geometric body - the polyhedron of a prism. 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, the bases of which are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure- straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture essential elements, of which the geometric body consists. These include:
Sometimes in geometry problems you can come across the concept of a section. The definition will sound like this: a section is all the points of a volumetric body belonging to a cutting plane. The section can be perpendicular (intersects 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 constructed - 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.
To find the reduced prismatic elements, use different relationships and formulas. Some of them are known from the planimetry course (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 its base and height:
V = Sbas h
Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in 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 development.
From the drawing it can be seen 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 = Posn h
Taking into account that the perimeter of the square is equal to P = 4a, the formula takes the form:
Sside = 4a h
For cube:
Sside = 4a²
To calculate the total surface area of the prism, you need to add 2 base areas to the lateral area:
Sfull = Sside + 2Smain
In relation to a quadrangular regular prism, the formula looks like:
Stotal = 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, the formulas can be derived:
- base side length: a = Sside / 4h = √(V / h);
- height or side rib length: h = Sside / 4a = V / a²;
- base area: Sbas = V / h;
- side face area: Side gr = Sside / 4.
To determine how much area the 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 a prism, use the formula:
dprize = √(2a² + h²)
To understand how to apply the given relationships, you can practice and solve several simple tasks.
Examples of problems with solutions
Here are some tasks found on 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 sand level be if you move it into a container of the same shape, but with a base twice as long?
It should be reasoned 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 denote the length of the base by 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, we can equate the expressions:
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 correct 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 at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, 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 a known diagonal:
a = d / √2 = 6√2 / √2 = 6
The total surface area is found using the formula for a 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, i.e. 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 area 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 involving a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and rectangle, as well as to know the formulas for finding the volume and surface area.
How to find the area of a cube
