A branch of mathematics that deals with the study of the properties of various figures (points, lines, angles, two-dimensional and three-dimensional objects), their sizes and relative positions. For ease of teaching, geometry is divided into planimetry and stereometry. IN… … Collier's Encyclopedia
Geometry of spaces of dimensions greater than three; the term is applied to those spaces whose geometry was originally developed for the case of three dimensions and only then generalized to the number of dimensions n>3, primarily Euclidean space, ... ... Mathematical Encyclopedia
N-dimensional Euclidean geometry is a generalization of Euclidean geometry to a space of more dimensions. Although physical space is three-dimensional, and human senses are designed to perceive three dimensions, N is dimensional... ... Wikipedia
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- (Constructive Solid Geometry, CSG) technology used in modeling solid bodies. Constructive block geometry is often, but not always, the way to model in 3D graphics and CAD. It allows you to create a complex scene or... Wikipedia
Constructive Solid Geometry (CSG) is a technology used in solid modeling. Constructive block geometry is often, but not always, the way to model in 3D graphics and CAD. She... ... Wikipedia
This term has other meanings, see Volume (meanings). Volume is an additive function of a set (a measure) characterizing the capacity of the area of space that it occupies. Initially arose and was applied without strict... ... Wikipedia
Cube Type Regular polyhedron Face square Vertices Edges Faces ... Wikipedia
Volume is an additive function of a set (a measure) characterizing the capacity of the area of space that it occupies. Initially it arose and was applied without a strict definition in relation to three-dimensional bodies of three-dimensional Euclidean space.... ... Wikipedia
A portion of space bounded by a collection of a finite number of planar polygons (see GEOMETRY) connected in such a way that each side of any polygon is a side of exactly one other polygon (called... ... Collier's Encyclopedia
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In the school curriculum for 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 is a straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture the most important elements that make up a geometric body. 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 (the maximum number of sections that can be constructed 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.
To find the reduced prismatic elements, various relations and formulas are used. 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, the new sand level 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
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"), like opposite sides of a 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 lateral faces equal to the number of 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.
How does she look
There are quite a lot of rectangular prisms in the environment of modern man. This is, for example, ordinary cardboard for shoes, computer components, etc. Look around. Even in a room you will probably see many rectangular prisms. This includes a computer case, a bookcase, a refrigerator, a wardrobe, and many other items. The shape is extremely popular mainly because it allows you to make the most of your space, whether you're decorating your interior or packing things into cardboard before moving.Properties of a rectangular prism
A rectangular prism has a number of specific properties. Any pair of faces can serve as it, since all adjacent faces are located at the same angle to each other, and this angle is 90°. The volume and surface area of a rectangular prism are easier to calculate than any other. Take any object that has the shape of a rectangular prism. Measure its length, width and height. To find the volume, just multiply these measurements. That is, the formula looks like this: V=a*b*h, where V is the volume, a and b are the sides of the base, h is the height that coincides with the side edge of this geometric body. The base area is calculated using the formula S1=a*b. For the side surface, you must first calculate the perimeter of the base using the formula P=2(a+b), and then multiply it by the height. The resulting formula is S2=P*h=2(a+b)*h. To calculate the total surface area of a rectangular prism, add twice the base area and the side surface area. The formula is S=2S1+S2=2*a*b+2*(a+b)*h=2Lecture: Prism, its bases, side ribs, height, lateral surface; straight prism; correct prism
Prism
If you learned flat figures with us from previous questions, then you are completely ready to study three-dimensional figures. The first solid we will learn will be a prism.
Prism is a three-dimensional body that has a large number of faces.
This figure has two polygons at the bases, which are located in parallel planes, and all the side faces have the shape of a parallelogram.
Fig. 1. Fig. 2
So, let's figure out what a prism consists of. To do this, pay attention to Fig. 1
As mentioned earlier, a prism has two bases that are parallel to each other - these are the pentagons ABCEF and GMNJK. Moreover, these polygons are equal to each other.
All other faces of the prism are called lateral faces - they consist of parallelograms. For example BMNC, AGKF, FKJE, etc.
The total surface of all lateral faces is called lateral surface.
Each pair of adjacent faces has a common side. This common side is called an edge. For example MV, SE, AB, etc.
If the upper and lower base of the prism are connected by a perpendicular, then it will be called the height of the prism. In the figure, the height is marked as straight line OO 1.
There are two main types of prism: oblique and straight.
If the lateral edges of the prism are not perpendicular to the bases, then such a prism is called inclined.
If all the edges of a prism are perpendicular to the bases, then such a prism is called straight.
If the bases of a prism contain regular polygons (those with equal sides), then such a prism is called correct.
If the bases of a prism are not parallel to each other, then such a prism will be called truncated.
You can see it in Fig. 2
Formulas for finding the volume and area of a prism
There are three basic formulas for finding volume. They differ from each other in application:
Similar formulas for finding the surface area of a prism:
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