What is the volume of the pyramid formula. How to find the volume of a pyramid

The word "pyramid" is involuntarily associated with the majestic giants in Egypt, faithfully keeping the peace of the pharaohs. Maybe that's why the pyramid is unmistakably recognized by everyone, even children.

However, let's try to give it a geometric definition. Let us imagine several points (A1, A2,..., An) on the plane and one more (E) that does not belong to it. So, if point E (top) is connected to the vertices of the polygon formed by points A1, A2, ..., Ap (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 and quadrangular, pentagonal, etc.

If you look closely at the pyramid, it will become clear why it is also defined differently - as geometric figure, which has a polygon at the base, and triangles united by a common vertex as side faces.

Since the pyramid is a spatial figure, then it also has such a quantitative characteristic, as it is calculated from the well-known equal third of the product of the base of the pyramid and its height:

The volume of the pyramid, when deriving the formula, is initially calculated for a triangular one, taking as a basis a constant ratio that relates this value to the volume of a triangular prism having the same base and height, which, as it turns out, is three times greater than this volume.

And since any pyramid is divided into triangular ones, and its volume does not depend on the constructions performed in the proof, the validity of the above volume formula is obvious.

Standing apart among all the pyramids are the right ones, in which the base lies. 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 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 very simply.

For example, to calculate the volume of a quadrangular pyramid, if it is regular, the length of the side of a regular quadrangle (square) at the base is squared and, multiplying 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 the ball inscribed in the pyramid and the area of its total surface;

as two thirds of the product of the distance between two arbitrarily taken crossing edges and the area of the parallelogram that forms the midpoints of the remaining four edges.

The volume of the pyramid is also 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 of pyramids, one cannot ignore the 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 volume of the pyramid, though not quite in it modern form, however, equal to 1/3 of the volume of the prism known to us, was found by Democritus. Archimedes called his counting method “without proof,” since Democritus approached the pyramid as if it were a figure made up of infinitely thin, similar plates.

Vector algebra also “addressed” the question of finding the volume of the pyramid, using the coordinates of its vertices for this. A pyramid built on a troika vectors a,b,c, is equal to one sixth of the modulus of the mixed product of the given vectors.

A pyramid is a polyhedron with a polygon at its base. All faces, in turn, form triangles that converge at one vertex. Pyramids are triangular, quadrangular, and so on. In order to determine which pyramid is in front of you, it is enough to count the number of corners at its base. The definition of "pyramid height" is very often found in geometry problems in school curriculum. In the article we will try to consider different ways her location.

Parts of the pyramid

Each pyramid consists of the following elements:

side faces that have three corners and converge at the top;
apothem represents the height that descends from its top;
the top of the pyramid is a point that connects the side edges, but does not lie in the plane of the base;
a base is a polygon that does not contain a vertex;
the height of the pyramid is a segment that intersects the top of the pyramid and forms a right angle with its base.

How to find the height of a pyramid if its volume is known

Through the formula V \u003d (S * h) / 3 (in the formula V is the volume, S is the base area, h is the height of the pyramid), we find that h \u003d (3 * V) / S. To consolidate the material, let's immediately solve the problem. The triangular base is 50 cm 2 while its volume is 125 cm 3 . The height of the triangular pyramid is unknown, which we need to find. Everything is simple here: we insert the data into our formula. We get h \u003d (3 * 125) / 50 \u003d 7.5 cm.

How to find the height of a pyramid if the length of the diagonal and its edge are known

As we remember, the height of the pyramid forms a right angle with its base. And this means that the height, edge and half of the diagonal together form Many, of course, remember the Pythagorean theorem. Knowing two dimensions, it will not be difficult to find the third value. Recall the well-known theorem a² = b² + c², where a is the hypotenuse, and in our case the edge of the pyramid; b - the first leg or half of the diagonal and c - respectively, the second leg, or the height of the pyramid. From this formula, c² = a² - b².

Now the problem: in a regular pyramid, the diagonal is 20 cm, while the length of the edge is 30 cm. You need to find the height. We solve: c² \u003d 30² - 20² \u003d 900-400 \u003d 500. Hence c \u003d √ 500 \u003d about 22.4.

How to find the height of a truncated pyramid

It is a polygon that has a section parallel to its base. The height of a truncated pyramid is the segment that connects its two bases. The height can be found at a regular pyramid if the lengths of the diagonals of both bases, as well as the edge of the pyramid, are known. Let the diagonal of the larger base be d1, while the diagonal of the smaller base is d2, and the edge has length l. To find the height, you can lower the heights from the two upper opposite points of the diagram to its base. We see that we have two right triangle, it remains to find the lengths of their legs. To do this, subtract the smaller diagonal from the larger diagonal and divide by 2. So we will find one leg: a \u003d (d1-d2) / 2. After that, according to the Pythagorean theorem, we only have to find the second leg, which is the height of the pyramid.

Now let's look at this whole thing in practice. We have a task ahead of us. The truncated pyramid has a square at the base, the diagonal length of the larger base is 10 cm, while the smaller one is 6 cm, and the edge is 4 cm. It is required to find the height. To begin with, we find one leg: a \u003d (10-6) / 2 \u003d 2 cm. One leg is 2 cm, and the hypotenuse is 4 cm. It turns out that the second leg or height will be 16-4 \u003d 12, that is, h \u003d √12 = about 3.5 cm.

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Lesson Objectives.

Educational: Derive a formula for calculating the volume of a pyramid

Developing: to develop students' cognitive interest in academic disciplines, the ability to apply their knowledge in practice.

Educational: to cultivate attention, accuracy, to expand 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, the base of the pyramid, ribs, height, axis, apothem. Which pyramid is called a regular, tetrahedron, truncated pyramid?

Pyramid - 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 summit pyramids, and a flat polygon is the base of the pyramid. Segments, connecting the top of the pyramid with the top 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 lies correct n-gon, a height base coincides with foundation center called correct n-gonal pyramid. axis A regular pyramid is called a straight line containing its height. A regular triangular pyramid is called a tetrahedron. If the pyramid is crossed by a plane parallel to the plane of the base, then it will cut off the pyramid, similar given. The rest 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. Complement this pyramid to a triangular prism with the same base and height.

3. This prism is composed of three pyramids:

1) this pyramid SABC.

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 the 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 coincident heights drawn from the 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 base area multiplied by the height.

3. Consolidation of new material. Solution of exercises.

1) Task № 33 from the textbook A.N. Pogorelov. Slides 7, 8, 9

On the side of the base? and side edge b find the volume of a regular pyramid, at the base of which lies:

1) triangle,

2) quadrilateral,

3) hexagon.

In a regular pyramid, the height passes through the center of a circle circumscribed near the base. Then: (Appendix)

4. Historical information about the pyramids. Slides 15, 16, 17

The first of our contemporaries who established a number of unusual phenomena associated with the pyramid was the French scientist Antoine Bovy. Exploring the pyramid of Cheops in the 30s of the twentieth century, he discovered that the bodies of small animals that accidentally got into the royal room were mummified. Bovi explained the reason for this for himself by the shape of the pyramid and, as it turned out, was not mistaken. His work formed the basis contemporary 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 be of practical importance.

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 form, or rather, in the form of an octahedron, half of which is the pyramid. The pyramid with the top up symbolizes life, the top down - death, other world. Just like the components of the Star of David (Magen David), where the triangle directed upwards symbolizes the ascent to the Higher Mind, God, and the triangle, lowered with its top down, symbolizes the descent of the soul to Earth, material existence ...

The digital value of the code by 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. In addition, the number 365 consists of three numbers 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. Therefore, 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 in general has a global significance: the structure of the human hands, in particular, is based on it, the total number of phalanges of the fingers of each of which is also 14. This code is also related to the constellation Ursa Major, which includes our Sun, and in which it once was another star that destroyed Phaeton, a planet located between Mars and Jupiter, after which Pluto appeared in the solar system, and the characteristics of the other planets changed.

Many esoteric sources claim that the humanity of the 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 the initiates. At the beginning of the cycles and half-cycles of the sidereal year, they built the pyramids. They came 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 knowledge of cosmic laws is embedded in the pyramids...

Special devices in the form of pyramids neutralize negative electromagnetic radiation on a person from a computer, TV, refrigerator and other household appliances.

In one of the books, a case is described when a pyramid installed in a car interior reduced fuel consumption and reduced the CO content in the exhaust gases.

Seeds of garden crops aged in pyramids had best germination and productivity. The publications even recommended soaking the seeds before sowing in pyramidal water.

It was found that the pyramids have a beneficial effect on the ecological situation. Eliminate pathogenic zones in apartments, offices and suburban areas, creating a positive aura.

Dutch researcher Paul Dickens in his book gives examples of the healing properties of the pyramids. He noticed that they can be used to relieve headaches, joint pain, stop bleeding with small cuts, and that the energy of the pyramids stimulates metabolism and strengthens the immune system.

In some modern publications, it is noted that medicines kept in the 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 alcohol, 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 need a high pyramid.

One newspaper article says that if you store jewelry under a pyramid, they self-cleanse and acquire a special shine, while precious and semi-precious stones accumulate positive bioenergy and then gradually give it away.

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 like their noble counterparts.

Perhaps this will not be relevant for many, but in a small pyramid, old razor blades are self-sharpening, 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 the energy of the pyramids.

Over the 5000 years of its existence, the pyramids have become a kind of symbol that personifies 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, publishing house “Enlightenment”.

3) Encyclopedia "Tree of Knowledge" Marshall K.

What is a pyramid?

How she looks like?

You see: at the pyramid below (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 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, quite "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;

At the same time, the point where it fell height, is called height base. Note that in the "crooked" pyramids height may even be outside the pyramid. Like this:

And there is nothing terrible in this. It looks like an obtuse triangle.

Correct pyramid.

Lots of difficult words? Let's decipher: " At the base - correct"- this is understandable. And now 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 right pyramid.

Hexagonal: at the base - a regular hexagon, the vertex is projected into the center of the base.

quadrangular: at the base - a square, the top is projected to the intersection point of the diagonals of this square.

triangular: at the base is a regular triangle, the vertex is projected to the intersection point of the heights (they are also medians and bisectors) of this triangle.

Very important properties of a regular pyramid:

In the right pyramid

all side edges are equal.
all side faces are isosceles triangles and all these triangles are equal.

Pyramid Volume

The main formula for the volume of the pyramid:

Where did it come from exactly? This is not so simple, and at first you just need to remember that the pyramid and cone have volume in the formula, but the 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. I need to find and.

This is the area right triangle.

Let's remember how to search for this area. We use the area formula:

We have "" - this, and "" - this too, eh.

Now let's find.

According to the Pythagorean theorem for

What does it matter? This is the radius of the circumscribed circle in, because pyramidcorrect and hence the center.

Since - the point of intersection and the median too.

(Pythagorean theorem for)

Substitute in the formula for.

Let's plug everything into the volume formula:

Attention: if you have a regular tetrahedron (i.e.), then the formula is:

Let the side of the base be equal, and the side edge equal.

There is no need to search here; because at the base is a square, and therefore.

Let's find. According to the Pythagorean theorem for

Do we know? Almost. Look:

(we saw this by reviewing).

Substitute in 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 searched for the area of a regular triangle when calculating the volume of a regular triangular pyramid, here we use the found formula.

Now let's find (this).

According to the Pythagorean theorem for

But what does it matter? It's simple because (and everyone else too) is correct.

We substitute:

\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))

PYRAMID. BRIEFLY ABOUT THE MAIN

A pyramid is a polyhedron that consists of any flat polygon (), a point that does not lie in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid to the 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, which has a regular polygon 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 side edges are equal.
All side faces are isosceles triangles and all these triangles are equal.

Volume of the pyramid:

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Do not know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

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And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

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