Find the total surface area of the prism. Theorem on the area of the lateral surface of a straight prism
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 faces) . 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 side 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.
Solution. 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.
Solution
.
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 A 1 of the upper base we lower the perpendicular to the plane of the lower base A 1 D. Its length will be the height of the prism. Consider D A 1 AD: since this is the angle of inclination of the side rib A 1 A to the base plane A 1 A= 8 cm. From this triangle we find A 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.
Solution. 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.
Solution. 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, because 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 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.
Different prisms are different from each other. At the same time, they have a lot in common. To find the area of \u200b\u200bthe base of a prism, you need to figure out what kind it looks like.
General theory
A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces - they can vary significantly in size.
When solving problems, it is not only the area of \u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.
Sometimes heights appear in tasks. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.
It should be noted that the area of the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.
triangular prism
It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.
Mathematical notation looks like this: S = ½ av.
To find out the area of \u200b\u200bthe base in a general form, the formulas are useful: Heron and the one in which half of the side is taken to the height drawn to it.
The first formula should be written like this: S \u003d √ (p (p-a) (p-in) (p-s)). This entry contains a semi-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you want to know the area of the base of a triangular prism, which is regular, then the triangle is equilateral. It has its own formula: S = ¼ a 2 * √3.
quadrangular prism
Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of \u200b\u200bthe base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = av, where a, b are the sides of the rectangle.
When it comes to a quadrangular prism, the base area of a regular prism is calculated using the formula for a square. Because it is he who lies at the base. S \u003d a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: na \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is na opposite to this angle.
If a rhombus lies at the base of the prism, then the same formula will be needed to determine its area as for a parallelogram (since it is a special case of it). But you can also use this one: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with a different number of vertices.
Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of \u200b\u200bthe base of the prism is equal to the area of one such triangle (the formula can be seen above), multiplied by five.
Regular hexagonal prism
According to the principle described for a pentagonal prism, it is possible to divide the base hexagon into 6 equilateral triangles. The formula for the area of the base of such a prism is similar to the previous one. Only in it should be multiplied by six.
The formula will look like this: S = 3/2 and 2 * √3.
Tasks
No. 1. A regular straight line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of \u200b\u200bthe base of the prism and the entire surface.
Solution. The base of a prism is a square, but its side is not known. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 \u003d d 2 - n 2. On the other hand, this segment "x" is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (d 2 - n 2) / 2.
Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.
To find out the area of \u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. The total surface area of the prism is found to be 960 cm 2 .
Answer. The base area of the prism is 144 cm2. The entire surface - 960 cm 2 .
No. 2. Dana At the base lies a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.
Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared times ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of one base of the prism.
All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of the side surface is wound 180 cm 2 .
Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 2.
Any polygon can lie at the base of the prism - a triangle, a quadrilateral, etc. Both bases are exactly the same, and accordingly, by which the angles of parallel faces are connected to each other, they are always parallel. At the base of a regular prism lies a regular polygon, that is, one in which all sides are equal. In a straight prism, the edges between the side faces are perpendicular to the base. In this case, a polygon with any number of angles can lie at the base of a straight prism. A prism whose base is a parallelogram is called a parallelepiped. A rectangle is a special case of a parallelogram. If this figure lies at the base, and the side faces are located at right angles to the base, the parallelepiped is called rectangular. The second name of this geometric body is rectangular.