The volume of an isosceles pyramid. Volume formulas for a regular triangular pyramid. Examples of problem solving

One of the simplest volumetric figures is a triangular pyramid, since it consists of the smallest number of faces from which a figure in space can be formed. In this article, we will consider the formulas with which you can find the volume of a triangular correct pyramid.

Triangular pyramid

According to general definition a pyramid is a polygon, all vertices of which are connected to one point not located in the plane of this polygon. If the latter is a triangle, then the whole figure is called a triangular pyramid.

The pyramid in question consists of a base (triangle) and three side faces (triangles). The point at which the three side faces are connected is called the top of the shape. The perpendicular dropped to the base from this top is the height of the pyramid. If the point of intersection of the perpendicular with the base coincides with the point of intersection of the medians of the triangle at the base, then they speak of a regular pyramid. Otherwise, it will be oblique.

As said, the foundation triangular pyramid may be a triangle general type... However, if it is equilateral, and the pyramid itself is straight, then they talk about the correct volumetric figure.

Any triangular pyramid has 4 faces, 6 edges, and 4 vertices. If the lengths of all edges are equal, then such a figure is called a tetrahedron.

The volume of a triangular pyramid of general type

Before writing down the formula for the volume of a regular triangular pyramid, we present the expression of this physical quantity for a general pyramid. This expression looks like:

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Here S o is the area of ​​the base, h is the height of the figure. This equality will be true for any type of base of a pyramid polygon, as well as for a cone. If at the base there is a triangle with the length of the side a and the height h o lowered onto it, then the formula for the volume will be written as follows:

V = 1/6 * a * h o * h.

Volume formulas for a regular triangular pyramid

A regular triangular pyramid has an equilateral triangle at its base. It is known that the height of this triangle is related to the length of its side by equality:

Substituting this expression into the formula for the volume of a triangular pyramid written in the previous paragraph, we get:

V = 1/6 * a * h o * h = √3 / 12 * a 2 * h.

The volume of a regular pyramid with a triangular base is a function of the length of the base side and the height of the figure.

Since anyone regular polygon can be inscribed in a circle, the radius of which will uniquely determine the length of the side of the polygon, then this formula can be written in terms of the corresponding radius r:

V = √3 / 4 * h * r 2.

This formula can be easily obtained from the previous one, if we take into account that the radius r of the circumscribed circle through the length of the side a of the triangle is determined by the expression:

The problem of determining the volume of a tetrahedron

Let us show how to use the above formulas in solving specific problems of geometry.

It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular pyramid-tetrahedron.

Recall that a tetrahedron is a regular triangular pyramid in which all bases are equal. To use the formula for the volume of a regular triangular pyramid, you need to calculate two quantities:

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• the length of the side of the triangle;
• figure height.

The first value is known from the condition of the problem:

To determine the height, consider the figure shown in the figure.

The marked triangle ABC is right-angled, where the angle ABC is 90 o. The AC side is the hypotenuse, the length of which is a. By simple geometric reasoning, one can show that the side BC has the length:

Note that the length BC is the radius of a circle circumscribed around a triangle.

h = AB = √ (AC 2 - BC 2) = √ (a 2 - a 2/3) = a * √ (2/3).

Now we can substitute h and a in the corresponding formula for volume:

V = √3 / 12 * a 2 * a * √ (2/3) = √2 / 12 * a 3.

Thus, we have obtained the formula for the volume of a tetrahedron. It can be seen that the volume depends only on the length of the rib. If we substitute the value from the condition of the problem into the expression, then we get the answer:

V = √2 / 12 * 7 3 ≈ 40.42 cm 3.

If we compare this value with the volume of a cube having the same edge, we get that the volume of a tetrahedron is 8.5 times less. This indicates that the tetrahedron is a compact figure that is realized in some natural substances. For example, a methane molecule is tetrahedral, and each carbon atom in a diamond is bonded to four other atoms to form a tetrahedron.

Problem with homothetic pyramids

Goals and objectives of the lesson:

4. Historical background 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. Exploring the pyramid of Cheops in the 30s of the twentieth century, he found that the bodies of small animals that accidentally got into the king's room were mummified. Bowie explained the reason for this for himself by the shape of the pyramid and, as it turned out, 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 be of practical importance.

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 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... In the same way as the constituent parts of the Star of David (Magen David), where the triangle directed upwards symbolizes the ascent to the Higher Reason, God, and the triangle, lowered with its apex downwards, symbolizes the descent of the soul to Earth, material existence ...

The digital value of the code that encrypts information about the Universe 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, 365 has 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 is Jupiter and 5 - Mars. Consequently, the Earth is in a special way connected 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 meaning: on it, in particular, the structure of human hands is based, total number phalanges of the fingers of each of which are also 14. This code has to do with the constellation Ursa Major, which includes our Sun, and in which there was once another star that destroyed Phaeton, a planet located between Mars and Jupiter, after which it appeared in the solar system Pluto, and the characteristics of the other planets have 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 passed on only to the initiates. At the beginning of the cycles and half cycles of the sidereal year, they built pyramids. They came close to discovering the code of life. The civilization of Atlantis succeeded a lot, 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 laid in the pyramids ...

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 when a pyramid installed in the passenger compartment of a car reduced fuel consumption and reduced the content of CO in exhaust gases.

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

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

The Dutch researcher Paul Dickens in his book gives examples of the healing properties of the pyramids. He noticed that with their help it is possible to relieve headaches, joint pains, to stop bleeding with 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 need a high pyramid.

One newspaper article tells you what if you store jewelry under the pyramid, they self-cleanse and acquire a special shine, and precious and semi-precious stones accumulate positive bioenergy and then gradually give it back.

According to American scientists, food products, such as cereals, flour, salt, sugar, coffee, tea, having visited the pyramid, improve their taste, and cheap cigarettes become similar to their noble counterparts.

Perhaps for many this will not be relevant, 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 evidence of the existence of the energy of the pyramids.

Over 5000 years of its existence, the pyramids have turned into a kind of symbol that personifies the human desire 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 "Education".

3) Encyclopedia "The Tree of Knowledge" Marshall K.

The main characteristic of any geometric figure in space is its volume. In this article, we will consider what a pyramid with a triangle at the base is, and 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 ancient Egyptian pyramids, however, they are rectangular regular, not triangular. Let's explain how to get a triangular pyramid.

Take an arbitrary triangle and connect all its vertices with some one point located outside the plane of this triangle. The formed figure will be called a triangular pyramid. It is shown in the figure below.

As you can see, the figure under consideration is formed by four triangles, which are generally different. Each triangle is a side or face of a pyramid. This pyramid is often called a tetrahedron, that is, a four-sided volumetric figure.

In addition to the sides, the pyramid also has edges (there are 6 of them) and vertices (there are 4 of them).

triangular base

A figure that is obtained using an arbitrary triangle and a point in space will generally be an irregular inclined pyramid. Now imagine that the original triangle has the same sides, and the point in space is located exactly above its geometric center at a distance h from the plane of the triangle. The pyramid built using this initial data will be correct.

Obviously, the number of edges, sides and vertices for a regular triangular pyramid will be the same as for a pyramid built from an arbitrary triangle.

but correct figure has some distinctive features:

• its height, drawn from the top, will exactly intersect the base in the geometric center (the point of intersection of the medians);
• side surface 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 form, for example, the crystal lattice of a diamond, where a carbon atom is connected to four of the same atoms by covalent bonds, or a methane molecule, where the tops of a 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 by the apothem h a lowered to this side, the formula for the volume of a triangular pyramid can be written in as follows:

V = 1/6 × a × h a × h

For a general type, determining the 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 look. The area of ​​the base (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 the same equilateral triangles. In this case, its volume can be determined only on the basis of knowledge of the parameter of its edge a. The corresponding expression is:

Truncated pyramid

If the upper part containing the vertex is cut off at a regular triangular pyramid, then you get a truncated figure. Unlike the original, 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, it is necessary 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 provided in the article clearer for the reader, we will show with an illustrative example how to use some of the written formulas.

Let the volume of the triangular pyramid be 15 cm 3. The figure is known to be correct. The apothem a b of the lateral rib should be found if it is known 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.

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 common in geometry problems in school curriculum... In the article, we will try to consider different ways finding it.

Parts of the pyramid

Each pyramid consists of the following elements:

• side faces, which have three corners and converge at the top;
• apothem is the height that descends from its top;
• the top of the pyramid is the point that connects side ribs, but at the same time does not lie in the plane of the base;
• base is a polygon that does not have a vertex;
• the height of the pyramid is a segment that crosses 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 = (S * h) / 3 (in the formula V is the volume, S is the area of ​​the base, h is the height of the pyramid), we find that h = (3 * V) / S. To consolidate the material, let's solve the problem right away. 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 data into our formula. We get h = (3 * 125) / 50 = 7.5 cm.

How to find the height of a pyramid if you know the length of the diagonal and its edges

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 measurements, it will not be difficult to find the third quantity. Let's remember 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 rib is 30 cm. It is necessary to find the height. We solve: c² = 30² - 20² = 900-400 = 500. Hence c = √ 500 = 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 a line segment that connects its two bases. The height can be found at the correct pyramid if the lengths of the diagonals of both bases are known, as well as the edge of the pyramid. 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 got two right triangle, it remains to find the lengths of their legs. To do this, subtract the smaller one from the larger diagonal and divide by 2. So we find one leg: a = (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 the whole thing in practice. We have a task before us. The truncated pyramid has a square at the base, the length of the diagonal 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 = (10-6) / 2 = 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 = 12, that is, h = √12 = about 3.5 cm.