The video course "Get an A" includes all the topics necessary for the successful passing of the exam in mathematics by 60-65 points. Completely all tasks 1-13 of the Profile USE in mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.

All the necessary theory. Quick solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solving complex problems of the 2nd part of the exam.

"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 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.

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

Insofar as 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

Prism. Parallelepiped

prism is called a polyhedron whose two faces are equal n-gons (grounds) , lying in parallel planes, and the remaining n faces are parallelograms (side edges) . Side rib prism is the side of the lateral face that does not belong to the base.

A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called oblique . Correct A prism is a straight prism whose bases are regular polygons.

Height prism is called the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. diagonal section A section of a prism by a plane passing through two side edges that do not belong to the same face is called. Perpendicular section called the section of the prism by a plane perpendicular to the lateral edge of the prism.

Side surface area prism is the sum of the areas of all side faces. Full surface area the sum of the areas of all the faces of the prism is called (i.e., the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism, the formulas are true:

where l is the length of the side rib;

H- height;

P

Q

S side

S full

S main is the area of ​​the bases;

V is the volume of the prism.

For a straight prism, the following formulas are true:

where p- the perimeter of the base;

l is the length of the side rib;

H- height.

Parallelepiped A prism whose base is a parallelogram is called. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called oblique . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped in which all edges are equal is called cube.

The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since the box is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of the parallelepiped intersect at one point and bisect it.

2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped, the following formulas are true:

where l is the length of the side rib;

H- height;

P is the perimeter of the perpendicular section;

Q– Area of ​​perpendicular section;

S side is the lateral surface area;

S full is the total surface area;

S main is the area of ​​the bases;

V is the volume of the prism.

For a right parallelepiped, the following formulas are true:

where p- the perimeter of the base;

l is the length of the side rib;

H is the height of the right parallelepiped.

For a rectangular parallelepiped, the following formulas are true:

(3)

where p- the perimeter of the base;

H- height;

d- diagonal;

a,b,c– measurements of a parallelepiped.

The correct formulas for a cube are:

where a is the length of the rib;

d is the diagonal of the cube.

Example 1 The diagonal of a rectangular cuboid is 33 dm, and its measurements are related as 2:6:9. Find the measurements of the cuboid.

Decision. To find the dimensions of the parallelepiped, we use formula (3), i.e. the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Denote by k coefficient of proportionality. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. We write formula (3) for the problem data:

Solving this equation for k, we get:

Hence, the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2 Find the volume of an inclined triangular prism whose base is an equilateral triangle with a side of 8 cm, if the lateral edge is equal to the side of the base and is inclined at an angle of 60º to the base.

Decision . Let's make a drawing (Fig. 3).

In order to find the volume of an inclined prism, you need to know the area of ​​\u200b\u200bits base and height. The area of ​​the base of this prism is the area of ​​an equilateral triangle with a side of 8 cm. Let's calculate it:

The height of a prism is the distance between its bases. From the top BUT 1 of the upper base we lower the perpendicular to the plane of the lower base BUT 1 D. Its length will be the height of the prism. Consider D BUT 1 AD: since this is the angle of inclination of the side rib BUT 1 BUT to the base plane BUT 1 BUT= 8 cm. From this triangle we find BUT 1 D:

Now we calculate the volume using formula (1):

Answer: 192 cm3.

Example 3 The lateral edge of a regular hexagonal prism is 14 cm. The area of ​​\u200b\u200bthe largest diagonal section is 168 cm 2. Find the total surface area of ​​the prism.

Decision. Let's make a drawing (Fig. 4)

The largest diagonal section is a rectangle AA 1 DD 1 , since the diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of ​​a prism, it is necessary to know the side of the base and the length of the lateral rib.

Knowing the area of ​​the diagonal section (rectangle), we find the diagonal of the base.

Because , then

Since then AB= 6 cm.

Then the perimeter of the base is:

Find the area of ​​the lateral surface of the prism:

The area of ​​a regular hexagon with a side of 6 cm is:

Find the total surface area of ​​the prism:

Answer:

Example 4 The base of a right parallelepiped is a rhombus. The areas of diagonal sections are 300 cm 2 and 875 cm 2. Find the area of ​​the side surface of the parallelepiped.

Decision. Let's make a drawing (Fig. 5).

Denote the side of the rhombus by a, the diagonals of the rhombus d 1 and d 2 , the height of the box h. To find the lateral surface area of ​​a straight parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, as ABCD- rhombus. H = AA 1 = h. That. Need to find a and h.

Consider diagonal sections. AA 1 SS 1 - a rectangle, one side of which is the diagonal of a rhombus AC = d 1 , second - side edge AA 1 = h, then

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we get the equality We get the following.

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 of ​​a regular prism can be expressed as follows:

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 area of ​​the lateral surface of this prism is equal to the sum of the areas of the three lateral 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).

Let's use the definition of the tangent in a 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.