S side surface of a straight prism. Everything You Need to Know About the Prism (2019)
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. Direct prism
Theorem 2. The area of the lateral surface of the prism
Parallelepiped :
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Dimensions 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 a polyhedron is called, 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 edges, the ends of the edges are called the tops of the prism. Lateral ribs called edges that do not belong to the bases. The union of side faces is called side surface of the prism, and the union of all faces is called full surface of the prism. Prism height 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 side edges are perpendicular to the planes of the bases. 
correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.
Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P ^ - perimeter of the perpendicular section;
S b - side surface area;
V - volume;
S p - area of the total surface of the prism.
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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 verify 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 holds for the triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to A "C") as the lines of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC equals A"C") as opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.
Definition 2 . A perpendicular section of a prismatic surface is a 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 the prismatic surface - side 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 to each other.
It is obvious that if the base of the prism ABCDE and one of the edges AA" are given in magnitude and direction, then it is possible to construct a prism by drawing the edges BB", CC", .., equal and parallel to the edge AA".
Definition 4 . The height of a prism is the distance between the planes of its bases (HH").
Definition 5
. A prism is called a straight line 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 edges 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 and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be the given prism and abcde be its perpendicular section, so that the segments ab, bc, .. are perpendicular to its side edges. Face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that matches ab; the area of \u200b\u200bthe face BCV "C" is equal to the product of the base BB" by the height bc, etc. Therefore, the side surface (i.e., the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", BB", .., by the sum ab+bc+cd+de+ea.
"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.
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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 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.
Decision. 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.
Decision. 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.
Polyhedra
The main object of study of stereometry are three-dimensional bodies. Body is a part of space bounded by some surface.
polyhedron A body whose surface consists of a finite number of plane polygons is called. A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices vertices of the polyhedron.
For example, a cube consists of six squares that are its faces. It contains 12 edges (sides of squares) and 8 vertices (vertices of squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
prism is called a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. The polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are side edges of the prism.
Prism height called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-coal if its base is an n-gon.
Any prism has the following properties, which follow from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The side edges of the prism are parallel and equal.
The surface of a prism is made up of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of the lateral surface of a prism is the sum of the areas of the lateral faces.
straight prism
The prism is called straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique.
The faces of a straight prism are rectangles. The height of a straight prism is equal to its side faces.
full prism surface is the sum of the lateral surface area and the areas of the bases.
Correct prism is called a right prism with a regular polygon at the base.
Theorem 13.1. The area of the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, equivalently, to the lateral edge).
Proof. The side faces of a straight prism are rectangles whose bases are the sides of the polygons at the bases of the prism, and the heights are the side edges of the prism. Then, by definition, the lateral surface area is:
,
where is the perimeter of the base of a straight prism.
Parallelepiped
If parallelograms lie at the bases of a prism, then it is called parallelepiped. All the faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.
Theorem 13.2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of the parallelepiped are parallelograms, then and , which means that according to T about two straight lines parallel to the third . In addition, this means that the lines and lie in the same plane (the plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals and intersect and the intersection point is divided in half, which was required to be proved.
A right parallelepiped whose base is a rectangle is called cuboid. All faces of a cuboid are rectangles. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). There are three sizes (width, height, length).
Theorem 13.3. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions 
(proved by applying Pythagorean T twice).
A rectangular parallelepiped in which all edges are equal is called cube.
Tasks
13.1 How many diagonals does n- carbon prism
13.2 In an inclined triangular prism, the distances between the side edges are 37, 13, and 40. Find the distance between the larger side face and the opposite side edge.
13.3 Through the side of the lower base of a regular triangular prism, a plane is drawn that intersects the side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.