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The video course “Get an A” includes all the topics necessary to successfully pass the Unified State Exam in mathematics with 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.
All the necessary theory. Quick solutions, pitfalls and secrets of the Unified State Exam. All current tasks of part 1 from the FIPI Task Bank have been analyzed. The course fully complies with the requirements of the Unified State Exam 2018.
The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.
Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. A basis for solving complex problems of Part 2 of the Unified State Exam.
Schoolchildren who are preparing to take the Unified State Exam in mathematics should definitely learn how to solve problems on finding the area of a straight and regular prism. Many years of practice confirm the fact that many students consider such geometry tasks to be quite difficult.
At the same time, high school students with any level of training should be able to find the area and volume of a regular and straight prism. Only in this case will they be able to count on receiving competitive scores based on the results of passing the Unified State Exam.
Key Points to Remember
- If the lateral edges of a prism are perpendicular to the base, it is called a straight line. All side faces of this figure are rectangles. The height of a straight prism coincides with its edge.
- A regular prism is one whose side edges are perpendicular to the base in which the regular polygon is located. The side faces of this figure are equal rectangles. A correct prism is always straight.
Preparing for the unified state exam together with Shkolkovo is the key to your success!
To make your classes easy and as effective as possible, choose our math portal. Here you will find all the necessary material that will help you prepare for passing the certification test.
Specialists of the Shkolkovo educational project propose to go from simple to complex: first we give theory, basic formulas, theorems and elementary problems with solutions, and then gradually move on to expert-level tasks.
Basic information is systematized and clearly presented in the “Theoretical Information” section. If you have already managed to repeat the necessary material, we recommend that you practice solving problems on finding the area and volume of a right prism. The “Catalog” section presents a large selection of exercises of varying degrees of difficulty.
Try to calculate the area of a straight and regular prism or right now. Analyze any task. If it does not cause any difficulties, you can safely move on to expert-level exercises. And if certain difficulties do arise, we recommend that you regularly prepare for the Unified State Exam online together with the Shkolkovo mathematical portal, and tasks on the topic “Straight and Regular Prism” will be easy for you.
What is the volume of a prism and how to find it
The volume of a prism is the product of the area of its base and its height.
However, we know that at the base of the prism there can be a triangle, a square or some other polyhedron.
Therefore, to find the volume of a prism, you simply need to calculate the area of the base of the prism, and then multiply this area by its height.
That is, if there is a triangle at the base of the prism, then first you need to find the area of the triangle. If the base of the prism is a square or other polygon, then first you need to look for the area of the square or other polygon.
It should be remembered that the height of the prism is the perpendicular drawn to the bases of the prism.
What is a prism
Now let's remember the definition of a prism.
A prism is a polygon, two faces (bases) of which are in parallel planes, and all edges located outside these faces are parallel.
To put it simply:
A prism is any geometric figure that has two equal bases and flat faces.
The name of a prism depends on the shape of its base. When the base of a prism is a triangle, then such a prism is called triangular. A polyhedral prism is a geometric figure whose base is a polyhedron. Also, a prism is a type of cylinder.
What types of prisms are there?
If we look at the picture above, we will see that prisms are straight, regular and oblique.
Exercise
1. Which prism is called correct?
2. Why is it called that?
3. What is the name of a prism whose bases are regular polygons?
4. What is the height of this figure?
5. What is a prism whose edges are not perpendicular called?
6. Define a triangular prism.
7. Can a prism be a parallelepiped?
8. What geometric figure is called a semiregular polygon?
What elements does a prism consist of?
A prism consists of elements such as a lower and upper base, side faces, edges and vertices.
Both bases of the prism lie in planes and are parallel to each other.
The side faces of the pyramid are parallelograms.
The lateral surface of a pyramid is the sum of its lateral faces.
The common sides of the side faces are nothing more than the side edges of a given figure.
The height of the pyramid is the segment connecting the planes of the bases and perpendicular to them.
Prism properties
A geometric figure, like a prism, has a number of properties. Let's take a closer look at these properties:
Firstly, the bases of a prism are equal polygons;
Secondly, the side faces of a prism are presented in the form of a parallelogram;
Thirdly, this geometric figure has parallel and equal edges;
Fourthly, the total surface area of the prism is:
Now let's look at the theorem, which provides the formula used to calculate the lateral surface area and proof.
Have you ever thought about such an interesting fact that not only a geometric body, but also other objects around us can be a prism? Even an ordinary snowflake, depending on the temperature, can turn into an ice prism, taking the shape of a hexagonal figure.
But calcite crystals have such a unique phenomenon as breaking up into fragments and taking on the shape of a parallelepiped. And what’s most amazing is that no matter how small the calcite crystals are crushed into, the result is always the same: they turn into tiny parallelepipeds.
It turns out that the prism has gained popularity not only in mathematics, demonstrating its geometric body, but also in the field of art, since it is the basis of paintings created by such great artists as P. Picasso, Braque, Griss and others.
In physics, a triangular prism made of glass is often used to study the spectrum of white light because it can resolve it into its individual components. In this article we will consider the volume formula
What is a triangular prism?
Before giving the volume formula, let's consider the properties of this figure.
To get this, you need to take a triangle of any shape and move it parallel to itself to some distance. The vertices of the triangle in the initial and final positions should be connected by straight segments. The resulting volumetric figure is called a triangular prism. It consists of five sides. Two of them are called bases: they are parallel and equal to each other. The bases of the prism in question are triangles. The three remaining sides are parallelograms.
In addition to the sides, the prism in question is characterized by six vertices (three for each base) and nine edges (6 edges lie in the planes of the bases and 3 edges are formed by the intersection of the sides). If the side edges are perpendicular to the bases, then such a prism is called rectangular.
The difference between a triangular prism and all other figures of this class is that it is always convex (four-, five-, ..., n-gonal prisms can also be concave).
This is a rectangular figure with an equilateral triangle at its base.
Volume of a general triangular prism
How to find the volume of a triangular prism? The formula in general is similar to that for a prism of any type. It has the following mathematical notation:
Here h is the height of the figure, that is, the distance between its bases, S o is the area of the triangle.
The value of S o can be found if some parameters for the triangle are known, for example, one side and two angles or two sides and one angle. The area of a triangle is equal to half the product of its height and the length of the side by which this height is lowered.
As for the height h of the figure, it is easiest to find it for a rectangular prism. In the latter case, h coincides with the length of the side edge.
Volume of a regular triangular prism
The general formula for the volume of a triangular prism, which is given in the previous section of the article, can be used to calculate the corresponding value for a regular triangular prism. Since its base is an equilateral triangle, its area is equal to:
Anyone can get this formula if they remember that in an equilateral triangle all angles are equal to each other and amount to 60 o. Here the symbol a is the length of the side of the triangle.
The height h is the length of the edge. It is in no way connected with the base of a regular prism and can take arbitrary values. As a result, the formula for the volume of a triangular prism of the correct type looks like this:
Having calculated the root, you can rewrite this formula as follows:
Thus, to find the volume of a regular prism with a triangular base, it is necessary to square the side of the base, multiply this value by the height and multiply the resulting value by 0.433.