Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Straight prism
Theorem 2. The area of the lateral surface of a prism
Parallelepiped:
Definition 6. Box
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped
Prism is called a polyhedron in which two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the ribs are called tops of the prism. Side ribs edges that do not belong to the bases are called. The union of side faces is called lateral surface of the prism, and the union of all faces is called full prism surface. The height of the prism is called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. 
Straight prism called a prism in which the lateral edges are perpendicular to the planes of the bases. 
Correct called a straight prism (Fig. 3), at the base of which a regular polygon lies.
Legend:
l - lateral rib;
P is the perimeter of the base;
S o - base area;
H - height;
P ^ - perimeter of the perpendicular section;
S b - lateral surface area;
V is the volume;
S p - the area of the full surface of the prism.
|
V = SH |
*It is assumed that every two consecutive planes intersect and that the last plane intersects the first
Theorem 1 ... Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A "B" C "D" E "be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A" B "C" are equal and have the same direction of rotation and that the same is true for triangles ABD and A "B" D ", ABE and A" B "E". But the corresponding sides of these triangles are parallel (for example, AC parallel to A "C") as lines of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example AC is equal to A "C") as opposite sides of the parallelogram and that the angles formed by these sides are equal and have the same direction.
Definition 2 ... The perpendicular section of a prismatic surface is called the section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.
Definition 3
... A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to a prismatic surface - side faces; edges of a prismatic surface - lateral edges of the prism... By virtue of the previous theorem, the bases of the prism are equal polygons... All side faces of the prism - parallelograms; all side edges are equal.
Obviously, if you are given the base of the prism ABCDE and one of the edges AA "in size and direction, then you can build a prism by drawing the edges BB", CC ", .., equal and parallel to the edge AA".
Definition 4 ... The height of the prism is the distance between the planes of its bases (HH ").
Definition 5
... A prism is called straight if its bases are perpendicular sections of a prismatic surface. In this case, the height of the prism is, of course, its side rib; side faces will rectangles.
Prisms can be classified by the number of side faces equal to the number of sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.
Theorem 2
... The area of the lateral surface of the prism is equal to the product of the lateral edge by the perimeter of the perpendicular section.
Let ABCDEA "B" C "D" E "- this prism and abcde - its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. Face ABA" B "is a parallelogram; its area is equal to the product of the base AA "to a height that coincides with ab; the area of the BCB "C" face is equal to the product of the base BB "by the height bc, etc. Therefore, the lateral surface (that is, the sum of the lateral face areas) is equal to the product of the lateral rib, in other words, the total length of the segments AA", BB ", .., for the amount ab + bc + cd + de + ea.
How she looks like
There are quite a few rectangular prisms surrounded by modern man. This is, for example, the usual cardboard from under shoes, computer components, etc. Look around. Even in a room, you will likely see many rectangular prisms. This is a computer case, a bookcase, a refrigerator, a wardrobe, and many other items. The shape is extremely popular mainly because it allows you to use space as efficiently as possible, regardless of whether you are decorating the interior or packing things in cardboard before moving.Rectangular prism properties
A rectangular prism has a number of specific properties. Any pair of faces can serve as it, since all adjacent faces are located to each other at the same angle, and this angle is 90 °. The volume and surface area of a rectangular prism is easier to calculate than any other. Take any object in the shape of a rectangular prism. Measure its length, width and height. To find the volume, it is enough to multiply these measurements. That is, the formula looks like this: V = a * b * h, where V is the volume, a and b are the sides of the base, h is the height, which for this geometric body coincides with the side edge. The base area is calculated using the formula S1 = a * b. For a side surface, you must first calculate the perimeter of the base using the formula P = 2 (a + b), and then multiply it by the height. It turns out the formula S2 = P * h = 2 (a + b) * h. Add twice the base area and the side area to calculate the total surface area of a rectangular prism. You get the formula S = 2S1 + S2 = 2 * a * b + 2 * (a + b) * h = 2Prism. 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 a prism is the side of the side face that does not belong to the base.
A prism whose side 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, the bases of which are regular polygons.
Height prism is called the distance between the planes of the bases. Diagonal prism is called a segment that connects two vertices that do not belong to the same face. Diagonal section the section of a prism is called a plane passing through two lateral edges that do not belong to one face. Perpendicular section the section of a prism is called a plane perpendicular to the lateral edge of the prism.
Side surface area prism is called the sum of the areas of all side faces. Full surface area called the sum of the areas of all faces of the prism (i.e. the sum of the areas of the side faces and the areas of the bases).
For an arbitrary prism, the following formulas are valid:
where l- the length of the side rib;
H- height;
P
Q
S side
S full
S main- the area of the bases;
V Is the volume of the prism.
For a straight prism, the following formulas are correct:
where p- base perimeter;
l- the length of the side rib;
H- height.
Parallelepiped called a prism, the base of which is a parallelogram. A parallelepiped with side edges 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 straight parallelepiped, the base of which is a rectangle, is called rectangular. A rectangular parallelepiped with all edges equal is called cube.
The faces of a parallelepiped that do not have common vertices are called opposing ... The lengths of the edges outgoing from one vertex are called measurements parallelepiped. Since a parallelepiped 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 are halved by it.
2. In a rectangular parallelepiped, the square of the diagonal length 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- the length of the side rib;
H- height;
P- the perimeter of the perpendicular section;
Q- The area of the perpendicular section;
S side- lateral surface area;
S full- total surface area;
S main- the area of the bases;
V Is the volume of the prism.
For a straight parallelepiped, the following formulas are true:
where p- base perimeter;
l- the length of the side rib;
H- the height of the straight parallelepiped.
For a rectangular parallelepiped, the following formulas are true:
(3)
where p- base perimeter;
H- height;
d- diagonal;
a, b, c- measurements of the parallelepiped.
For a cube, the following formulas are correct:
where a- rib length;
d Is the diagonal of the cube.
Example 1. The diagonal of a rectangular parallelepiped is 33 dm, and its dimensions are related as 2: 6: 9. Find the dimensions of the parallelepiped.
Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. by the fact that the square of the hypotenuse of a rectangular parallelepiped is equal to the sum of the squares of its dimensions. Let us denote by k proportionality coefficient. Then the dimensions of the parallelepiped will be 2 k, 6k and 9 k... Let's write the formula (3) for the problem data:
Solving this equation for k, we get:
This means that 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, the base of which 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, it is necessary to know its base area 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 plane of the base, A 1 A= 8 cm.From this triangle we find A 1 D:
Now we calculate the volume by the formula (1):
Answer: 192 cm 3.
Example 3. The lateral edge of a regular hexagonal prism is 14 cm. The area of the largest diagonal section is 168 cm 2. Find the total surface area of the prism.
Solution. Let's make a drawing (fig. 4)
Largest Diagonal Section - Rectangle AA 1 DD 1, since the diagonal AD regular hexagon ABCDEF is the greatest. In order to calculate the area of the lateral surface of the 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.
Since, then 
Since then AB= 6 cm.
Then the perimeter of the base is:
Let us find the area of the lateral surface of the prism:
The area of a regular hexagon with a side of 6 cm is equal to:
Find the total surface area of the prism:
Answer: 
Example 4. The base of the rectangle is a rhombus. The areas of the diagonal sections are 300 cm 2 and 875 cm 2. Find the area of the side surface of a parallelepiped.
Solution. Let's make a drawing (fig. 5).
Let us denote the side of the rhombus through a, the diagonals of the rhombus d 1 and d 2, the height of the parallelepiped h... To find the area of the lateral surface of a straight parallelepiped, 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 - rectangle, one side of which is the diagonal of the rhombus AS = d 1, the second is a lateral rib 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 obtain the equality. We obtain the following.
Different prisms are not alike. At the same time, they have a lot in common. To find the area of the base of a prism, you need to figure out what kind it has.
General theory
A prism is any polyhedron, the sides of which are in 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. That does not apply to the side faces - they can vary significantly in size.
When solving problems, not only the area of the base of the prism is encountered. Knowledge of the lateral surface, that is, all faces that are not bases, may be required. The full surface will already be the union of all the faces that make up the prism.
Sometimes the tasks include height. 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 base area of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same shapes at the top and bottom edges, then their areas will be equal.
Triangular prism
It has at its base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to remember that its area is determined by half the product of the legs.
The mathematical notation looks like this: S = ½ av.
To find out the area of the base in 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 = √ (p (p-a) (p-c) (p-c)). 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 turns out to be equilateral. There is a formula for it: S = ¼ a 2 * √3.
Quadrangular prism
Its base is any of the known quadrangles. It can be a rectangle or square, parallelepiped or rhombus. In each case, in order to calculate the area of the base of the prism, you will need a different formula.
If the base is a rectangle, then its area is determined as follows: S = ab, 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 turns out to be at the bottom. S = a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S = a * na. It happens that the side of the parallelepiped and one of the corners are given. Then, to calculate the height, you will need to use an additional formula: n a = b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is n a opposite to this angle.
If there is a rhombus at the base of the prism, then the same formula will be needed to determine its area as for the parallelogram (since it is its special case). But you can also use this: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves dividing 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 the 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 base area 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
№ 1. Given a correct straight line. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of the base of the prism and the entire surface.
Solution. The base of the 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 = d 2 - n 2. On the other hand, this segment "x" is a hypotenuse in a triangle, the legs of which are equal to the side of the square. That is, x 2 = a 2 + a 2. Thus, it turns out that a 2 = (d 2 - n 2) / 2.
Substitute 22 instead of d, and replace "n" with its value - 14, then it turns out that the side of the square is 12 cm. Now just find out the area of the base: 12 * 12 = 144 cm 2.
To find out the area of the entire surface, you need to add twice the base area and quadruple the side. The latter can be easily found using 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 960 cm 2.
Answer. The base area of the prism is 144 cm 2. The entire surface is 960 cm 2.
№ 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: base and side surface.
Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area is equal to 6 squared, multiplied by 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 there are exactly so many side faces of the prism. Then the lateral surface area turns out to be 180 cm 2 wound.
Answer. Areas: base - 9√3 cm 2, lateral surface of the prism - 180 cm 2.
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Books
- A set of tables. Geometry. Grade 10. 14 tables + methodology,. The tables are printed on thick polygraphic cardboard 680 x 980 mm in size. The kit includes a brochure with guidelines for teachers. Educational album of 14 sheets. ...