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 the area of the base in general view, 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 (r-a) (r-in) (r-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 \u200b\u200bthe base of a triangular prism, which is regular, then the triangle turns out to be 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 identical 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.
"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 the trapezoid ABCD. Proof. Showing pictures. Proof of the theorem.
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With the help of this video tutorial, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area. During the lesson, the teacher will explain what these geometric figures, as a polyhedron and prisms, will give the appropriate definitions and explain their essence on concrete examples.
With the help of this lesson, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area.
Definition. A surface composed of polygons and bounding a certain geometric body will be called a polyhedral surface or a polyhedron.
Consider the following examples of polyhedra:
1. Tetrahedron ABCD is a surface made up of four triangles: ABC, ADB, bdc and ADC(Fig. 1).
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2. Parallelepiped ABCDA 1 B 1 C 1 D 1 is a surface composed of six parallelograms (Fig. 2).
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The main elements of a polyhedron are faces, edges, vertices.
The faces are the polygons that make up the polyhedron.
Edges are sides of faces.
The vertices are the ends of the edges.
Consider a tetrahedron ABCD(Fig. 1). Let us indicate its main elements.
Facets: triangles ABC, ADB, BDC, ADC.
Ribs: AB, AC, BC, DC, AD, BD.
Peaks: A, B, C, D.
Consider a box ABCDA 1 B 1 C 1 D 1(Fig. 2).
Facets: parallelograms AA 1 D 1 D, D 1 DCC 1, BB 1 C 1 C, AA 1 B 1 B, ABCD, A 1 B 1 C 1 D 1 .
Ribs: AA 1 , BB 1 , SS 1 , DD 1 , AD, A 1 D 1 , B 1 C 1 , BC, AB, A 1 B 1 , D 1 C 1 , DC.
Peaks: A, B, C, D, A 1 ,B 1 ,C 1 ,D 1 .
An important special case of a polyhedron is a prism.
ABSA 1 IN 1 WITH 1(Fig. 3).
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Equal Triangles ABC and A 1 B 1 C 1 are located in parallel planes α and β so that the edges AA 1 , BB 1 , SS 1 are parallel.
That is ABSA 1 IN 1 WITH 1- triangular prism, if:
1) Triangles ABC and A 1 B 1 C 1 are equal.
2) Triangles ABC and A 1 B 1 C 1 located in parallel planes α and β: ABC║A 1 B 1 C (α ║ β).
3) Ribs AA 1 , BB 1 , SS 1 are parallel.
ABC and A 1 B 1 C 1- the base of the prism.
AA 1 , BB 1 , SS 1- side ribs of the prism.
If from an arbitrary point H 1 one plane (for example, β) drop the perpendicular HH 1 onto the plane α, then this perpendicular is called the height of the prism.
Definition. If the lateral edges are perpendicular to the bases, then the prism is called straight, otherwise it is called oblique.
Consider a triangular prism ABSA 1 IN 1 WITH 1(Fig. 4). This prism is straight. That is, its side edges are perpendicular to the bases.
For example, rib AA 1 perpendicular to the plane ABC. Edge AA 1 is the height of this prism.
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Note that the side face AA 1 V 1 V perpendicular to the bases ABC and A 1 B 1 C 1, since it passes through the perpendicular AA 1 to the foundations.
Now consider an inclined prism ABSA 1 IN 1 WITH 1(Fig. 5). Here the lateral edge is not perpendicular to the plane of the base. If we drop from the point A 1 perpendicular A 1 H on the ABC, then this perpendicular will be the height of the prism. Note that the segment AN is the projection of the segment AA 1 to the plane ABC.
Then the angle between the line AA 1 and plane ABC is the angle between the line AA 1 and her AN projection onto a plane, that is, the angle A 1 AH.
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Consider a quadrangular prism ABCDA 1 B 1 C 1 D 1(Fig. 6). Let's see how it turns out.
1) Quadrilateral ABCD equal to a quadrilateral A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.
2) Quadrangles ABCD and A 1 B 1 C 1 D 1 ABC║A 1 B 1 C (α ║ β).
3) Quadrangles ABCD and A 1 B 1 C 1 D 1 arranged so that the lateral ribs are parallel, that is: AA 1 ║BB 1 ║SS 1 ║DD 1.
Definition. The diagonal of a prism is a segment connecting two vertices of a prism that do not belong to the same face.
For example, AC 1- diagonal of a quadrangular prism ABCDA 1 B 1 C 1 D 1.
Definition. If the side edge AA 1 perpendicular to the plane of the base, then such a prism is called a straight line.
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A special case of a quadrangular prism is the known parallelepiped. Parallelepiped ABCDA 1 B 1 C 1 D 1 shown in fig. 7.
Let's see how it works:
1) Equal figures lie in the bases. In this case - equal parallelograms ABCD and A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.
2) Parallelograms ABCD and A 1 B 1 C 1 D 1 lie in parallel planes α and β: ABC║A 1 B 1 C 1 (α ║ β).
3) Parallelograms ABCD and A 1 B 1 C 1 D 1 arranged in such a way that the side ribs are parallel to each other: AA 1 ║BB 1 ║SS 1 ║DD 1.
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From a point A 1 drop the perpendicular AN to the plane ABC. Line segment A 1 H is the height.
Let's take a look at how it works hexagonal prism(Fig. 8).
1) Equal hexagons lie at the base ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1: ABCDEF= A 1 B 1 C 1 D 1 E 1 F 1.
2) Planes of hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 parallel, that is, the bases lie in parallel planes: ABC║A 1 B 1 C (α ║ β).
3) Hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 arranged so that all side edges are parallel to each other: AA 1 ║BB 1 …║FF 1.
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Definition. If any side edge is perpendicular to the plane of the base, then such a hexagonal prism is called a straight line.
Definition. A right prism is called regular if its bases are regular polygons.
Consider a regular triangular prism ABSA 1 IN 1 WITH 1.
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triangular prism ABSA 1 IN 1 WITH 1- correct, this means that regular triangles lie at the bases, that is, all sides of these triangles are equal. Also, this prism is straight. This means that the side edge is perpendicular to the plane of the base. And this means that all side faces are equal rectangles.
So if a triangular prism ABSA 1 IN 1 WITH 1 is correct, then:
1) The side edge is perpendicular to the plane of the base, that is, it is the height: AA 1 ⊥ ABC.
2) The base is a regular triangle: ∆ ABC- right.
Definition. area full surface A prism is the sum of the areas of all its faces. Denoted S full.
Definition. The area of the lateral surface is the sum of the areas of all lateral faces. Denoted S side.
The prism has two bases. Then the total surface area of the prism is:
S full \u003d S side + 2S main.
The area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
The proof will be carried out on the example of a triangular prism.
Given: ABSA 1 IN 1 WITH 1- direct prism, i.e. AA 1 ⊥ ABC.
AA 1 = h.
Prove: S side \u003d R main ∙ h.
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Proof.
triangular prism ABSA 1 IN 1 WITH 1- straight, so AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C - rectangles.
Find the area of the lateral surface as the sum of the areas of the rectangles AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C:
S side \u003d AB ∙ h + BC ∙ h + CA ∙ h \u003d (AB + BC + CA) ∙ h \u003d P main ∙ h.
We get S side \u003d R main ∙ h, Q.E.D.
We got acquainted with polyhedrons, a prism, its varieties. We proved the theorem on the lateral surface of a prism. In the next lesson, we will solve problems on a prism.
- Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M .: Mnemosyne, 2008. - 288 p. : ill.
- Geometry. Grade 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general educational institutions with in-depth and specialized study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M. : Bustard, 008. - 233 p. :ill.
- Iclass().
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- What is the minimum number of faces a prism can have? How many vertices, edges does such a prism have?
- Is there a prism that has exactly 100 edges?
- The side rib is inclined to the base plane at an angle of 60°. Find the height of the prism if the side edge is 6 cm.
- In a straight line triangular prism all edges are equal. Its lateral surface area is 27 cm 2 . Find the total surface area of the prism.
Definition. Prism- this is a polyhedron, all the vertices of which are located in two parallel planes, and in the same two planes there are two faces of the prism, which are equal polygons with respectively parallel sides, and all edges that do not lie in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form side surface of the prism .
All side faces of the prism are parallelograms .
Edges that do not lie at the bases are called lateral edges of the prism ( AA 1, B.B. 1, CC 1, DD 1, EE 1).
Prism Diagonal a segment is called, the ends of which are two vertices of the prism that do not lie on one of its faces (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in the order of the bypass, the vertices of one base are indicated, and then, in the same order, the vertices of the other; the ends of each side edge are designated by the same letters, only the vertices lying in one base are indicated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1, a pentagon lies at the base, so the prism is called pentagonal prism. But since such a prism has 7 faces, then it heptahedron(2 faces are the bases of the prism, 5 faces are parallelograms, are its side faces)
Among straight prisms stands out private view: regular prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped- This is a quadrangular prism, at the base of which lies a parallelogram (oblique parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.cuboid - a right parallelepiped, the base of which is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar known properties parallelogram. A rectangular parallelepiped having equal measurements, are called cube
.A cube has all faces equal squares. The square of a diagonal is equal to the sum of the squares of its three dimensions 
,
where d is the diagonal of the square;
a - side of the square.
The idea of a prism is given by:
- various architectural structures;
- Kids toys;
- packing boxes;
- designer items etc.
Total and lateral surface area of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its side faces. the bases of the prism are equal polygons, then their areas are equal. That's whyS full \u003d S side + 2S main,
where S full- total surface area, S side- side surface area, S main- base area
The area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side\u003d P main * h,
where S side is the area of the lateral surface of a straight prism,
P main - the perimeter of the base of a straight prism,
h is the height of a straight prism, equal to side rib.
Prism Volume
The volume of a prism is equal to the product of the area of the base and the height.
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