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 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 hexahedron whose bases are 2 squares, and side faces represented by rectangles. Another name for this geometric figure- 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. 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 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
The area of the lateral surface of the prism. Hello! In this publication, we will analyze a group of tasks on stereometry. Consider a combination of bodies - a prism and a cylinder. At the moment, this article completes the entire series of articles related to the consideration of types of tasks in stereometry.
If new tasks appear in the task bank, then, of course, there will be additions to the blog in the future. But what is already there is quite enough so that you can learn how to solve all problems with a short answer as part of the exam. The material will be enough for years to come (the program in mathematics is static).
The presented tasks are related to the calculation of the area of the prism. I note that below we consider a straight prism (and, accordingly, a straight cylinder).
Without knowing any formulas, we understand that the lateral surface of a prism is all its lateral faces. In a straight prism, the side faces are rectangles.
The lateral surface area of such a prism is equal to the sum of the areas of all its lateral faces (that is, rectangles). If we are talking about a regular prism in which a cylinder is inscribed, then it is clear that all the faces of this prism are EQUAL rectangles.
Formally, the lateral surface area right prism can be expressed like this:
27064. A regular quadrangular prism is circumscribed about a cylinder whose base radius and height are equal to 1. Find the area of the lateral surface of the prism.
The lateral surface of this prism consists of four rectangles equal in area. The height of the face is 1, the edge of the base of the prism is 2 (these are two radii of the cylinder), so the area of the side face is:
Side surface area:
73023. Find the area of the lateral surface of a regular triangular prism circumscribed about a cylinder whose base radius is √0.12 and whose height is 3.
The lateral surface area of a given prism is equal to the sum three side faces (rectangles). To find the area of the side face, you need to know its height and the length of the base edge. The height is three. Find the length of the edge of the base. Consider the projection (top view):
We have a regular triangle in which a circle with radius √0.12 is inscribed. From the right triangle AOC we can find AC. And then AD (AD=2AC). By definition of tangent:
So AD \u003d 2AC \u003d 1.2. Thus, the area of \u200b\u200bthe lateral surface is equal to:
27066. Find the area of the lateral surface of a regular hexagonal prism circumscribed about a cylinder whose base radius is √75 and whose height is 1.
The desired area is equal to the sum of the areas of all side faces. For a regular hexagonal prism, the side faces are equal rectangles.
To find the area of a face, you need to know its height and the length of the base edge. The height is known, it is equal to 1.
Find the length of the edge of the base. Consider the projection (top view):
We have a regular hexagon in which a circle of radius √75 is inscribed.
Consider a right triangle ABO. We know the leg OB (this is the radius of the cylinder). we can also determine the angle AOB, it is equal to 300 (triangle AOC is equilateral, OB is a bisector).
We use the definition of tangent in right triangle:
AC \u003d 2AB, since OB is a median, that is, it divides AC in half, which means AC \u003d 10.
Thus, the area of the side face is 1∙10=10 and the area of the side surface is:
76485. Find the area of the lateral surface of a regular triangular prism inscribed in a cylinder whose base radius is 8√3 and whose height is 6.
The area of the lateral surface of the specified prism of three equal-sized faces (rectangles). To find the area, you need to know the length of the edge of the base of the prism (we know the height). If we consider the projection (top view), then we have a regular triangle inscribed in a circle. The side of this triangle is expressed in terms of the radius as:
Details of this relationship. So it will be equal
Then the area of the side face is equal to: 24∙6=144. And the required area:
245354. A regular quadrangular prism is circumscribed near a cylinder whose base radius is 2. The lateral surface area of the prism is 48. Find the height of the cylinder.
Everything is simple. We have four side faces equal in area, hence the area of one face is 48:4=12. Since the radius of the base of the cylinder is 2, then the edge of the base of the prism will be early 4 - it is equal to the diameter of the cylinder (these are two radii). We know the area of the face and one edge, the second being the height will be equal to 12:4=3.
27065. Find the area of the lateral surface of a regular triangular prism circumscribed about a cylinder whose base radius is √3 and whose height is 2.
Sincerely, Alexander.
Polyhedra
The main object of study of stereometry are three-dimensional bodies. Body is a part of space bounded by some surface.
polyhedron A body whose surface consists of a finite number of plane polygons is called. A polyhedron is called convex if it lies on one side of the plane of every flat 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 vertices of the polyhedron.
For example, a cube consists of six squares that are its faces. It contains 12 edges (sides of squares) and 8 vertices (vertices of squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
prism is called 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. The polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are side edges of the prism.
Prism height 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-coal if its base is an n-gon.
Any prism has the following properties, which follow from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The side edges of the prism are parallel and equal.
The surface of a prism is made up 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 side edges are perpendicular to the bases. Otherwise, the prism is called oblique.
The faces of a straight prism are rectangles. The height of a straight prism is equal to its side faces.
full prism surface is the sum of the lateral surface area and the areas of the bases.
Correct prism called a straight prism regular polygon at the 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, equivalently, to the lateral edge).
Proof. The side faces of a straight prism are rectangles whose bases are the sides of the polygons at the bases of the prism, and the heights are the side 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 the 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 the parallelepiped intersect at one point and the intersection point is divided in half.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of the parallelepiped are parallelograms, then and , which means that according to T about two straight lines parallel to the third . In addition, this means that the lines and lie in the same plane (the 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 and intersect and the intersection point is divided in half, which was to be proved.
A right parallelepiped whose base is a rectangle is called cuboid. All faces of a cuboid are rectangles. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). There are three sizes (width, height, length).
Theorem 13.3. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions 
(proved by applying Pythagorean T twice).
cuboid, in which all edges are equal, is called cube.
Tasks
13.1 How many diagonals does 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 face and the opposite side edge.
13.3 Through the side of the lower base of a regular triangular prism, a plane is drawn that intersects the side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.
Definition. Prism- this is a polyhedron, all the vertices of which are located in two parallel planes, and in the same two planes there are two faces of the prism, which are equal polygons with, respectively parallel sides, and all edges not lying in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form side surface prisms .
All side faces of a prism are parallelograms .
Edges that do not lie at the bases are called lateral edges of the prism ( AA 1, B.B. 1, CC 1, DD 1, EE 1).
Prism Diagonal a segment is called, the ends of which are two vertices of the prism that do not lie on one of its faces (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in the order of the bypass, the vertices of one base are indicated, and then, in the same order, the vertices of the other; the ends of each side edge are indicated by the same letters, only the vertices lying in one base are indicated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1, the base is a pentagon, so the prism is called pentagonal prism. But since such a prism has 7 faces, then it heptahedron(2 faces are the bases of the prism, 5 faces are parallelograms, are its side faces)
Among straight prisms, a particular type stands out: regular prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped- This is a quadrangular prism, at the base of which lies a parallelogram (oblique parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.cuboid- a right parallelepiped whose base is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar to the well-known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube
.A cube has all faces equal squares. The square of a diagonal is equal to the sum of the squares of its three dimensions 
,
where d is the diagonal of the square;
a - side of the square.
The idea of a prism is given by:
- various architectural structures;
- Kids toys;
- packing boxes;
- designer items, etc.
Total and lateral surface area of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its side faces. the bases of the prism are equal polygons, then their areas are equal. That's whyS full \u003d S side + 2S main,
Where S full- total surface area, S side- side surface area, S main- base area
The area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side\u003d P main * h,
Where S side is the area of the lateral surface of a straight prism,
P main - the perimeter of the base of a straight prism,
h is the height of a straight prism, equal to side rib.
Prism Volume
The volume of a prism is equal to the product of the area of the base and the height.
"Lesson of the Pythagorean theorem" - The Pythagorean theorem. Determine the type of quadrilateral KMNP. Warm up. Introduction to the theorem. Determine the type of triangle: Lesson plan: Historical digression. Solving simple problems. And find a ladder 125 feet long. Calculate the height CF of trapezoid ABCD. Proof. Showing pictures. Proof of the theorem.
"Volume of a prism" - The concept of a prism. direct 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 lines triangular prisms with height h. Draw the altitude of triangle ABC. The solution of the problem. Lesson goals. Basic steps in proving the direct prism theorem? Study of the prism volume theorem.
"Prism polyhedra" - Define a polyhedron. DABC is a tetrahedron, a convex polyhedron. The use of prisms. Where are prisms used? ABCDMP is an octahedron, made up of eight triangles. ABCDA1B1C1D1 is a parallelepiped, a convex polyhedron. Convex polyhedron. The concept of a polyhedron. Polyhedron A1A2..AnB1B2..Bn is a prism.
"Prism class 10" - A prism is a polyhedron whose faces are in parallel planes. The use of a prism in everyday life. Sside = Pbased. + h For a straight prism: Sp.p = Pmain. h + 2Smain. Inclined. Correct. Straight. Prism. Formulas for finding the area. The use of prism in architecture. Sp.p \u003d S side + 2 S based.
"Proof of the Pythagorean theorem" - Geometric proof. The meaning of the Pythagorean theorem. Pythagorean theorem. Euclid's proof. "In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs." Proofs 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.
