The word “pyramid” is involuntarily associated with the majestic giants in Egypt, faithfully guarding the peace of the pharaohs. Maybe that’s why everyone, even children, recognizes the pyramid unmistakably.
Nevertheless, let's try to give it a geometric definition. Let us imagine several points on the plane (A1, A2,..., An) and one more (E) that does not belong to it. So, if point E (vertex) is connected to the vertices of the polygon formed by points A1, A2,..., An (base), you get a polyhedron, which is called a pyramid. Obviously, the polygon at the base of the pyramid can have any number of vertices, and depending on their number, the pyramid can be called triangular, quadrangular, pentagonal, etc.
If you look closely at the pyramid, it will become clear why it is also defined in another way - as a geometric figure with a polygon at its base, and triangles united by a common vertex as its side faces.
Since the pyramid is a spatial figure, it also has the following quantitative characteristic, as calculated from the well-known equal third of the product of the base of the pyramid and its height:
When deriving the formula, the volume of a pyramid is initially calculated for a triangular one, taking as a basis a constant ratio connecting this value with the volume triangular prism, having the same base and height, which, as it turns out, is three times this volume.
And since any pyramid is divided into triangular ones, and its volume does not depend on the constructions performed during the proof, the validity of the given volume formula is obvious.
Standing apart among all the pyramids are the correct ones, which have at their base As for, it should “end” in the center of the base.
In the case of an irregular polygon at the base, to calculate the area of the base you will need:
- break it into triangles and squares;
- calculate the area of each of them;
- add up the received data.
In the case of a regular polygon at the base of the pyramid, its area is calculated using ready-made formulas, so the volume of a regular pyramid is calculated quite simply.
For example, to calculate the volume of a quadrangular pyramid, if it is regular, the length of the side of a regular quadrilateral (square) at the base is squared and, multiplied by the height of the pyramid, the resulting product is divided by three.
The volume of the pyramid can be calculated using other parameters:
- as a third of the product of the radius of a ball inscribed in a pyramid and its total surface area;
- as two-thirds of the product of the distance between two arbitrarily chosen crossing edges and the area of the parallelogram that forms the midpoints of the remaining four edges.
The volume of a pyramid is calculated simply in the case when its height coincides with one of the side edges, that is, in the case of a rectangular pyramid.
Speaking about pyramids, we cannot ignore truncated pyramids, obtained by cutting the pyramid with a plane parallel to the base. Their volume is almost equal to the difference between the volumes of the whole pyramid and the cut off top.
The first is the volume of the pyramid, although not entirely in its modern form, however, equal to 1/3 of the volume of the prism known to us, Democritus found. Archimedes called his method of calculation “without proof,” since Democritus approached the pyramid as a figure composed of infinitely thin, similar plates.
Vector algebra also “addressed” the issue of finding the volume of a pyramid, using the coordinates of its vertices. Pyramid built on three vectors a,b,c, is equal to one sixth of the modulus of the mixed product of the given vectors.
What is a pyramid?
How does she look?
You see: at the bottom of the pyramid (they say “ at the base") some polygon, and all the vertices of this polygon are connected to some point in space (this point is called " vertex»).
This whole structure still has side faces, side ribs And base ribs. Once again, let's draw a pyramid along with all these names:
Some pyramids may look very strange, but they are still pyramids.
Here, for example, is completely “oblique” pyramid.
And a little more about the names: if there is a triangle at the base of the pyramid, then the pyramid is called triangular, if it is a quadrangle, then quadrangular, and if it is a centagon, then... guess for yourself.
At the same time, the point where it fell height, called height base. Please note that in the “crooked” pyramids height may even end up outside the pyramid. Like this:
And there’s nothing wrong with that. It looks like an obtuse triangle.
Correct pyramid.
A lot of complex words? Let's decipher: “At the base - correct” - this is understandable. Now let us remember that a regular polygon has a center - a point that is the center of and , and .
Well, the words “the top is projected into the center of the base” mean that the base of the height falls exactly into the center of the base. Look how smooth and cute it looks regular pyramid.
Hexagonal: at the base there is a regular hexagon, the vertex is projected into the center of the base.
Quadrangular: the base is a square, the top is projected to the point of intersection of the diagonals of this square.
Triangular: at the base there is a regular triangle, the vertex is projected to the point of intersection of the heights (they are also medians and bisectors) of this triangle.
Very important properties of a regular pyramid:
- all side edges are equal.
- all side faces - isosceles triangles and all these triangles are equal.
Volume of the pyramid
The main formula for the volume of a pyramid:
Where exactly did it come from? This is not so simple, and at first you just need to remember that a pyramid and a cone have volume in the formula, but a cylinder does not.
Now let's calculate the volume of the most popular pyramids.
Let the side of the base be equal and the side edge equal. We need to find and.
This is the area of a regular triangle.
Let's remember how to look for this area. We use the area formula:
For us, “ ” is this, and “ ” is also this, eh.
Now let's find it.
According to the Pythagorean theorem for
What's the difference? This is the circumradius in because pyramidcorrect and, therefore, the center.
Since - the point of intersection of the medians too.
(Pythagorean theorem for)
Let's substitute it into the formula for.
And let’s substitute everything into the volume formula:
Attention: if you have a regular tetrahedron (i.e.), then the formula turns out like this:
Let the side of the base be equal and the side edge equal.
There is no need to look here; After all, the base is a square, and therefore.
We'll find it. According to the Pythagorean theorem for
Do we know? Almost. Look:
(we saw this by looking at it).
Substitute into the formula for:
And now we substitute and into the volume formula.
Let the side of the base be equal and the side edge.
How to find? Look, a hexagon consists of exactly six identical regular triangles. We have already looked for the area of a regular triangle when calculating the volume of a regular triangle. triangular pyramid, here we use the found formula.
Now let's find (it).
According to the Pythagorean theorem for
But what does it matter? It's simple because (and everyone else too) is correct.
Let's substitute:
\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))
PYRAMID. BRIEFLY ABOUT THE MAIN THINGS
A pyramid is a polyhedron that consists of any flat polygon (), a point not lying in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid with points of the base (side edges).
A perpendicular dropped from the top of the pyramid to the plane of the base.
Correct pyramid- a pyramid in which a regular polygon lies at the base, and the top of the pyramid is projected into the center of the base.
Property of a regular pyramid:
- In a regular pyramid, all lateral edges are equal.
- All lateral faces are isosceles triangles and all these triangles are equal.
Pyramid volume:
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Lesson Objectives.
Educational: Derive a formula for calculating the volume of a pyramid
Developmental: to develop students’ cognitive interest in academic disciplines, the ability to apply their knowledge in practice.
Educational: cultivate attention, accuracy, broaden the horizons of students.
Equipment and materials: computer, screen, projector, presentation “Volume of the Pyramid.”
1. Frontal survey. Slides 2, 3
What is called a pyramid, base of the pyramid, ribs, height, axis, apothem. Which pyramid is called regular, tetrahedron, truncated pyramid?
A pyramid is a polyhedron consisting of a flat polygon, points, not lying in the plane of this polygon and all segments, connecting this point with the points of the polygon.
This point called top pyramids, and a flat polygon is the base of the pyramid. Segments connecting the top of the pyramid with the vertices of the base are called ribs . Height pyramids - perpendicular, lowered from the top of the pyramid to the plane of the base. Apothem - side edge height correct pyramid. The pyramid, which at the base is correct n-gon, A height base coincides with center of the base called correct n-gonal pyramid. Axis of a regular pyramid is the straight line containing its height. A regular triangular pyramid is called a tetrahedron. If the pyramid is intersected by a plane, parallel to the plane base, then it will cut off the pyramid, similar given. The remaining part is called truncated pyramid.
2. Derivation of the formula for calculating the volume of the pyramid V=SH/3 Slides 4, 5, 6
1. Let SABC be a triangular pyramid with vertex S and base ABC.
2. Let's add this pyramid to a triangular prism with the same base and height.
3. This prism is composed of three pyramids:
1) of this SABC pyramid.
2) pyramids SCC 1 B 1.
3) and pyramids SCBB 1.
4. The second and third pyramids have equal bases CC 1 B 1 and B 1 BC and a total height drawn from the vertex S to the face of the parallelogram BB 1 C 1 C. Therefore, they have equal volumes.
5. The first and third pyramids also have equal bases SAB and BB 1 S and coinciding heights drawn from vertex C to the face of the parallelogram ABB 1 S. Therefore, they also have equal volumes.
This means that all three pyramids have the same volume. Since the sum of these volumes is equal to the volume of the prism, the volumes of the pyramids are equal to SH/3.
The volume of any triangular pyramid is equal to one third of the product of the area of the base and the height.
3. Consolidation of new material. Solution of exercises.
1) Problem № 33 from the textbook by A.N. Pogorelova. Slides 7, 8, 9
On the base side? and side edge b, find the volume of a regular pyramid, the base of which lies:
1) triangle,
2) quadrangle,
3) hexagon.
In a regular pyramid, the height passes through the center of a circle circumscribed around the base. Then: (Appendix)
4. Historical information about the pyramids. Slides 15, 16, 17
The first of our contemporaries to establish a number of unusual phenomena associated with the pyramid was the French scientist Antoine Bovy. While exploring the Cheops pyramid in the 30s of the twentieth century, he discovered that the bodies of small animals that accidentally ended up in the royal room were mummified. Bovey explained the reason for this to himself by the shape of a pyramid and, as it turned out, he was not mistaken. His works formed the basis modern research, as a result of which, over the past 20 years, many books and publications have appeared confirming that the energy of the pyramids can have practical significance.
The Mystery of the Pyramids
Some researchers argue that the pyramid contains a huge amount of information about the structure of the Universe, the solar system and man, encoded in its geometric shape, or more precisely, in the shape of an octahedron, half of which the pyramid represents. The pyramid with its top up symbolizes life, with its top down – death, the other world. Just like the components of the Star of David (Magen David), where the triangle directed upward symbolizes the ascent to the Higher Mind, God, and the triangle with its apex downward symbolizes the descent of the soul to Earth, material existence...
The digital value of the code with which information about the Universe is encrypted in the pyramid, the number 365, was not chosen by chance. First of all, this is the annual life cycle of our planet. Also, the number 365 is made up of three digits 3, 6 and 5. What do they mean? If in solar system The Sun passes at number 1, Mercury - 2, Venus - 3, Earth - 4, Mars - 5, Jupiter - 6, Saturn - 7, Uranus - 8, Neptune - 9, Pluto - 10, then 3 is Venus, 6 - Jupiter and 5 – Mars. Consequently, the Earth is connected in a special way with these planets. Adding the numbers 3, 6 and 5, we get 14, of which 1 is the Sun, and 4 is the Earth.
The number 14 generally has global significance: in particular, the structure of human hands is based on it, total number phalanges of the fingers of each of which are also 14. This code is also related to the constellation Ursa Major, which includes our Sun, and in which there was once another star that destroyed Phaethon, a planet located between Mars and Jupiter, after which it appeared in the solar system Pluto, and the characteristics of the other planets changed.
Many esoteric sources claim that humanity on Earth has already experienced a worldwide catastrophe four times. The third Lemurian race knew the Divine science of the Universe, then this secret doctrine was transmitted only to initiates. At the beginning of the cycles and half-cycles of the sidereal year, they built pyramids. They were close to discovering the code of life. The civilization of Atlantis succeeded in many things, but at some level of knowledge they were stopped by another planetary catastrophe, accompanied by a change of races. Probably, the initiates wanted to convey to us that the pyramids contain knowledge of cosmic laws...
Special devices in the form of pyramids neutralize negative electromagnetic radiation on a person from a computer, TV, refrigerator and other electrical appliances.
One of the books describes a case where a pyramid installed in the passenger compartment of a car reduced fuel consumption and reduced the CO content in exhaust gases.
Seeds of garden crops kept in pyramids had better germination and yield. Publications even recommended soaking the seeds in pyramid water before sowing.
Pyramids have been found to have beneficial effects on environmental situation. Eliminate pathogenic zones in apartments, offices and summer cottages, creating a positive aura.
Dutch researcher Paul Dickens in his book gives examples of the healing properties of the pyramids. He noticed that with their help you can relieve headaches, joint pain, stop bleeding from small cuts, and that the energy of the pyramids stimulates metabolism and strengthens the immune system.
Some modern publications note that medicines kept in a pyramid shorten the course of treatment, and the dressing material, saturated with positive energy, promotes wound healing.
Cosmetic creams and ointments improve their effect.
Drinks, including alcoholic ones, improve their taste, and the water contained in 40% vodka becomes healing. True, in order to charge a standard 0.5 liter bottle with positive energy, you will need a high pyramid.
One newspaper article says that if you store jewelry under the pyramid they self-clean and acquire a special shine, and precious and semi-precious stones accumulate positive bioenergy and then gradually release it.
According to American scientists, food products, such as cereals, flour, salt, sugar, coffee, tea, after being in the pyramid, improve their taste, and cheap cigarettes become similar to their noble brothers.
This may not be relevant for many, but in a small pyramid old razor blades sharpen themselves, and in a large pyramid water does not freeze at -40 degrees Celsius.
According to most researchers, all this is proof of the existence of pyramid energy.
Over the 5000 years of its existence, the pyramids have become a kind of symbol, personifying the desire of man to reach the pinnacle of knowledge.
5. Summing up the lesson.
Bibliography.
1) http://schools.techno.ru
2) Pogorelov A.V. Geometry 10-11, Prosveshchenie publishing house.
3) Encyclopedia “Tree of Knowledge” Marshall K.
Theorem.
The volume of the pyramid is equal to one third of the product of the area of the base and the height.
Proof:
First we prove the theorem for a triangular pyramid, then for an arbitrary one.1. Consider a triangular pyramidOABCwith volume V, base areaS and height h. Let's draw the axis oh (OM2- height), consider the sectionA1 B1 C1pyramid with a plane perpendicular to the axisOhand, therefore, parallel to the plane of the base. Let us denote byX abscissa point M1 intersection of this plane with the x axis, and throughS(x)- cross-sectional area. Let's express S(x) through S, h And X. Note that triangles A1 IN1 WITH1 And ABCs are similar. Indeed A1 IN1 II AB, so triangle OA 1 IN 1 similar to triangle OAB. WITH therefore, A1 IN1 : AB= OA 1: OA .
Right Triangles OA 1 IN 1 and OAV are also similar (they have a common acute angle with vertex O). Therefore, OA 1: OA = O 1 M1 : OM = x: h. Thus A 1 IN 1 : A B = x: h.Similarly, it is proved thatB1 C1:Sun = X: h And A1 C1:AC = X: h.So, triangleA1 B1 C1 And ABCsimilar with similarity coefficient X: h.Therefore, S(x) : S = (x: h)², or S(x) = S x²/ h².
Let us now apply the basic formula for calculating the volumes of bodies ata= 0, b =h we get
Thus, the volume of the original pyramid is 1/3Sh. The theorem has been proven.
Consequence:
Volume V of a truncated pyramid whose height is h and whose base areas are S and S1 , are calculated by the formula
h - height of the pyramid
S top - area of the upper base
S lower - area of the lower base
The main characteristic of any geometric figure in space is its volume. In this article we will look at what a pyramid with a triangle at the base is, and we will also show how to find the volume of a triangular pyramid - regular full and truncated.
What is this - a triangular pyramid?
Everyone has heard of the ancients Egyptian pyramids, however, they are regular quadrangular, not triangular. Let's explain how to get a triangular pyramid.
Let's take an arbitrary triangle and connect all its vertices with some single point located outside the plane of this triangle. The resulting figure will be called a triangular pyramid. It is shown in the figure below.
As you can see, the figure in question is formed by four triangles, which general case are different. Each triangle is the sides of the pyramid or its face. This pyramid is often called a tetrahedron, that is, a tetrahedral three-dimensional figure.
In addition to the sides, the pyramid also has edges (there are 6 of them) and vertices (of 4).
with triangular base
A figure that is obtained using an arbitrary triangle and a point in space will be an irregular slanted pyramid in the general case. Now imagine that the original triangle has identical sides, and a point in space is located exactly above its geometric center at a distance h from the plane of the triangle. The pyramid constructed using these initial data will be correct.
Obviously, the number of edges, sides and vertices of a regular triangular pyramid will be the same as that of a pyramid built from an arbitrary triangle.
However correct figure has some distinctive features:
- its height drawn from the vertex will exactly intersect the base at the geometric center (the point of intersection of the medians);
- the lateral surface of such a pyramid is formed by three identical triangles, which are isosceles or equilateral.
A regular triangular pyramid is not only a purely theoretical geometric object. Some structures in nature have its shape, for example crystal cell diamond, where a carbon atom is connected to four similar atoms covalent bonds, or methane molecule, where the tops of the pyramid are formed by hydrogen atoms.
triangular pyramid
You can determine the volume of absolutely any pyramid with an arbitrary n-gon at the base using the following expression:
Here the symbol S o denotes the area of the base, h is the height of the figure drawn to the marked base from the top of the pyramid.
Since the area of an arbitrary triangle is equal to half the product of the length of its side a and the apothem h a dropped onto this side, the formula for the volume of a triangular pyramid can be written in the following form:
V = 1/6 × a × h a × h
For general type Determining height is not an easy task. To solve it, the easiest way is to use the formula for the distance between a point (vertex) and a plane (triangular base), represented by the equation general view.
For the correct one, it has a specific appearance. The area of the base (of an equilateral triangle) for it is equal to:
Substituting it into the general expression for V, we get:
V = √3/12 × a 2 × h
A special case is the situation when all sides of a tetrahedron turn out to be identical equilateral triangles. In this case, its volume can be determined only based on knowledge of the parameter of its edge a. The corresponding expression looks like:
Truncated pyramid
If top part, containing the vertex, cut off from a regular triangular pyramid, you get a truncated figure. Unlike the original one, it will consist of two equilateral triangular bases and three isosceles trapezoids.
The photo below shows what a regular truncated triangular pyramid made of paper looks like.
To determine the volume of a truncated triangular pyramid, you need to know its three linear characteristics: each of the sides of the bases and the height of the figure, equal to the distance between the upper and lower bases. The corresponding formula for volume is written as follows:
V = √3/12 × h × (A 2 + a 2 + A × a)
Here h is the height of the figure, A and a are the lengths of the sides of the large (lower) and small (upper) equilateral triangles, respectively.
The solution of the problem
To make the information in the article clearer to the reader, we will show with a clear example how to use some of the written formulas.
Let the volume of the triangular pyramid be 15 cm 3 . It is known that the figure is correct. You should find the apothem a b of the lateral edge if you know that the height of the pyramid is 4 cm.
Since the volume and height of the figure are known, you can use the appropriate formula to calculate the length of the side of its base. We have:
V = √3/12 × a 2 × h =>
a = 12 × V / (√3 × h) = 12 × 15 / (√3 × 4) = 25.98 cm
a b = √(h 2 + a 2 / 12) = √(16 + 25.98 2 / 12) = 8.5 cm
The calculated length of the apothem of the figure turned out to be greater than its height, which is true for any type of pyramid.