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.
Measure all required distances in meters. The volume of many three-dimensional figures is easy to calculate using the appropriate formulas. However, all values substituted into the formulas must be measured in meters. Thus, before substituting values into the formula, make sure that they are all measured in meters, or that you have converted other units of measurement to meters.
- 1 mm = 0.001 m
- 1 cm = 0.01 m
- 1 km = 1000 m
To calculate the volume of rectangular shapes (rectangular box, cube) use the formula: volume = L × W × H(length times width times height). This formula can be considered as the product of the surface area of one of the faces of the figure and the edge perpendicular to this face.
- For example, let's calculate the volume of a room with a length of 4 m, a width of 3 m and a height of 2.5 m. To do this, simply multiply the length by the width by the height:
- 4×3×2.5
- = 12 × 2.5
- = 30. The volume of this room is 30 m 3.
- A cube is a three-dimensional figure in which all sides are equal. Thus, the formula for calculating the volume of a cube can be written as: volume \u003d L 3 (or W 3, or H 3).
To calculate the volume of figures in the form of a cylinder, use the formula: pi× R 2 × H. The calculation of the volume of a cylinder is reduced to multiplying the area of the round base by the height (or length) of the cylinder. Find the area of the circular base by multiplying pi (3.14) by the square of the circle's radius (R) (the radius is the distance from the center of the circle to any point on that circle). Then multiply the result by the height of the cylinder (H) and you will find the volume of the cylinder. All values are measured in meters.
- For example, let's calculate the volume of a well with a diameter of 1.5 m and a depth of 10 m. Divide the diameter by 2 to get the radius: 1.5/2=0.75 m.
- (3.14) × 0.75 2 × 10
- = (3.14) × 0.5625 × 10
- = 17.66. The volume of the well is 17.66 m3.
To calculate the volume of a sphere, use the formula: 4/3 x pi× R 3 . That is, you only need to know the radius (R) of the ball.
- For example, let's calculate the volume of a balloon with a diameter of 10 m. Divide the diameter by 2 to get the radius: 10/2=5 m.
- 4/3 x pi × (5) 3
- = 4/3 x (3.14) x 125
- = 4.189 × 125
- = 523.6. The volume of the balloon is 523.6 m 3.
To calculate the volume of figures in the form of a cone, use the formula: 1/3 x pi× R 2 × H. The volume of a cone is 1/3 of the volume of a cylinder that has the same height and radius.
- For example, let's calculate the volume of an ice cream cone with a radius of 3 cm and a height of 15 cm. Converting to meters, we get: 0.03 m and 0.15 m, respectively.
- 1/3 x (3.14) x 0.03 2 x 0.15
- = 1/3 x (3.14) x 0.0009 x 0.15
- = 1/3 × 0.0004239
- = 0.000141. The volume of an ice cream cone is 0.000141 m 3.
Use several formulas to calculate the volume of irregular shapes. To do this, try to break the figure into several shapes of the correct shape. Then find the volume of each such figure and add up the results.
- For example, let's calculate the volume of a small granary. The storage has a cylindrical body 12 m high and a radius of 1.5 m. The storage also has a conical roof 1 m high. By calculating the volume of the roof and the volume of the body separately, we can find the total volume of the granary:
- pi × R 2 × H + 1/3 x pi × R 2 × H
- (3.14) x 1.5 2 x 12 + 1/3 x (3.14) x 1.5 2 x 1
- = (3.14) × 2.25 × 12 + 1/3 x (3.14) × 2.25 × 1
- = (3.14) × 27 + 1/3 x (3.14) × 2.25
- = 84,822 + 2,356
- = 87.178. The volume of the granary is 87.178 m3.
Any geometric body can be characterized by surface area (S) and volume (V). Area and volume are not the same thing. An object can have a relatively small V and a large S, for example, this is how the human brain works. It is much easier to calculate these indicators for simple geometric shapes.
Parallelepiped: definition, types and properties
A parallelepiped is a quadrangular prism with a parallelogram at its base. Why might you need a formula for finding the volume of a figure? Books, packing boxes and many other things from everyday life have a similar shape. Rooms in residential and office buildings, as a rule, are rectangular parallelepipeds. To install ventilation, air conditioning and determine the number of heating elements in a room, it is necessary to calculate the volume of the room.
The figure has 6 faces - parallelograms and 12 edges, two arbitrarily chosen faces are called bases. The parallelepiped can be of several types. The differences are due to the angles between adjacent edges. The formulas for finding the V-s of various polygons are slightly different.
If 6 faces of a geometric figure are rectangles, then it is also called rectangular. A cube is a special case of a parallelepiped in which all 6 faces are equal squares. In this case, to find V, you need to know the length of only one side and raise it to the third power.
To solve problems, you will need knowledge not only of ready-made formulas, but of the properties of the figure. The list of basic properties of a rectangular prism is small and very easy to understand:
- Opposite faces of the figure are equal and parallel. This means that the ribs located opposite are the same in length and angle of inclination.
- All side faces of a right parallelepiped are rectangles.
- The four main diagonals of a geometric figure intersect at one point, and divide it in half.
- The square of the diagonal of a parallelepiped is equal to the sum of the squares of the dimensions of the figure (follows from the Pythagorean theorem).
Pythagorean theorem states that the sum of the areas of the squares built on the legs of a right triangle is equal to the area of the triangle built on the hypotenuse of the same triangle.
The proof of the last property can be seen in the image below. The course of solving the problem is simple and does not require detailed explanations.
The formula for the volume of a rectangular parallelepiped
The formula for finding for all types of geometric shapes is the same: V=S*h, where V is the desired volume, S is the area of the base of the parallelepiped, h is the height lowered from the opposite vertex and perpendicular to the base. In a rectangle, h coincides with one of the sides of the figure, so to find the volume of a rectangular prism, you need to multiply three measurements.
The volume is usually expressed in cm3. Knowing all three values a, b and c, finding the volume of the figure is not at all difficult. The most common type of problem in the USE is the search for the volume or diagonal of a parallelepiped. It is impossible to solve many typical USE tasks without a formula for the volume of a rectangle. An example of a task and the design of its solution is shown in the figure below.
Note 1. The surface area of a rectangular prism can be found by multiplying by 2 the sum of the areas of the three faces of the figure: the base (ab) and two adjacent side faces (bc + ac).
Note 2. The surface area of the side faces can be easily found by multiplying the perimeter of the base by the height of the parallelepiped.
Based on the first property of parallelepipeds, AB = A1B1, and the face B1D1 = BD. According to the consequences of the Pythagorean theorem, the sum of all angles in a right triangle is equal to 180 °, and the leg opposite the angle of 30 ° is equal to the hypotenuse. Applying this knowledge for a triangle, we can easily find the length of the sides AB and AD. Then we multiply the obtained values and calculate the volume of the parallelepiped.
The formula for finding the volume of a slanted box
To find the volume of an inclined parallelepiped, it is necessary to multiply the area of \u200b\u200bthe base of the figure by the height lowered to this base from the opposite angle.
Thus, the desired V can be represented as h - the number of sheets with an area S of the base, so the volume of the deck is made up of the Vs of all cards.
Examples of problem solving
The tasks of the single exam must be completed within a certain time. Typical tasks, as a rule, do not contain a large number of calculations and complex fractions. Often a student is offered how to find the volume of an irregular geometric figure. In such cases, you should remember the simple rule that the total volume is equal to the sum of the V-s of the constituent parts.
As you can see from the example in the image above, there is nothing complicated in solving such problems. Tasks from more complex sections require knowledge of the Pythagorean theorem and its consequences, as well as the formula for the length of the diagonal of a figure. To successfully solve test tasks, it is enough to familiarize yourself with samples of typical tasks in advance.
General review. Formulas of stereometry!
Hello dear friends! In this article, I decided to make a general overview of the problems in stereometry, which will be USE in mathematics e. It must be said that the tasks from this group are quite diverse, but not difficult. These are tasks for finding geometric quantities: lengths, angles, areas, volumes.
Considered: a cube, a rectangular parallelepiped, a prism, a pyramid, a compound polyhedron, a cylinder, a cone, a ball. It is sad that some graduates do not even take on such tasks at the exam itself, although more than 50% of them are solved elementarily, almost verbally.
The rest require little effort, knowledge and special techniques. In future articles, we will consider these tasks, do not miss it, subscribe to the blog update.
To solve, you need to know surface area and volume formulas parallelepiped, pyramid, prism, cylinder, cone and sphere. There are no complex tasks, they are all solved in 2-3 steps, it is important to "see" what formula needs to be applied.
All necessary formulas are presented below:
Ball or sphere. A spherical or spherical surface (sometimes simply a sphere) is the locus of points in space that are equidistant from one point - the center of the ball.
Ball volume equal to the volume of the pyramid, the base of which has the same area as the surface of the ball, and the height is the radius of the ball
The volume of a sphere is one and a half times less than the volume of a cylinder circumscribed around it.
A round cone can be obtained by rotating a right triangle around one of its legs, so a round cone is also called a cone of revolution. See also Surface area of a circular cone
Volume of a round cone is equal to one third of the product of the base area S and the height H:
(H - cube edge height)
A parallelepiped is a prism whose base is a parallelogram. The parallelepiped has six faces, and all of them are parallelograms. A parallelepiped whose four lateral faces are rectangles is called a right parallelepiped. A right box in which all six faces are rectangles is called a rectangular box.
Volume of a cuboid is equal to the product of the area of the base and the height:
(S is the area of the base of the pyramid, h is the height of the pyramid)
A pyramid is a polyhedron with one face - the base of the pyramid - an arbitrary polygon, and the rest - side faces - triangles with a common vertex, called the top of the pyramid.
A section parallel to the base of the pyramid divides the pyramid into two parts. The part of the pyramid between its base and this section is a truncated pyramid.
Volume of a truncated pyramid is equal to one third of the product of the height h (OS) by the sum of the areas of the upper base S1 (abcde), the lower base of the truncated pyramid S2 (ABCD) and the average proportional between them.
| 1. | V= |
n - the number of sides of a regular polygon - the bases of a regular pyramid
a - side of regular polygon - bases of regular pyramid
h - the height of the regular pyramid
A regular triangular pyramid is a polyhedron with one face - the base of the pyramid - a regular triangle, and the rest - side faces - equal triangles with a common vertex. The height descends to the center of the base from the top.
Volume of a regular triangular pyramid is equal to one third of the product of the area of an equilateral triangle, which is the base S (ABC) to the height h (OS)
a - side of a regular triangle - bases of a regular triangular pyramid
h - the height of a regular triangular pyramid
Derivation of the formula for the volume of a tetrahedron
The volume of a tetrahedron is calculated using the classical formula for the volume of a pyramid. It is necessary to substitute the height of the tetrahedron and the area of a regular (equilateral) triangle into it.
Volume of a tetrahedron- is equal to the fraction in the numerator of which the square root of two in the denominator is twelve, multiplied by the cube of the length of the edge of the tetrahedron
(h is the length of the side of the rhombus)
Circumference p is about three whole and one seventh the length of the diameter of a circle. The exact ratio of the circumference of a circle to its diameter is denoted by the Greek letter π
As a result, the perimeter of a circle or the circumference of a circle is calculated by the formula
| π rn |
(r is the radius of the arc, n is the central angle of the arc in degrees.)
