Calculate the volume of a triangular pyramid. The volume of the quadrangular pyramid
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.
However, the 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);
- 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 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, dropped to 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 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 a general equation.
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 three of its 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 the school curriculum. In the article we will try to consider different ways of 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 a point that connects the side edges, but 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 base area, 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. 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 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-angled triangles, 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.
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 regular pyramid.
Triangular pyramid
According to the general definition, a pyramid is a polygon, all vertices of which are connected to one point that is 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 mentioned, the base of a triangular pyramid can be a general triangle. 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.
general type
Before writing down a regular triangular pyramid, we give an expression for this physical quantity for a general pyramid. This expression looks like:
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:
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 any 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:
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 the tetrahedron is regular in which all bases are equal. To use the triangular volume formula, you need to calculate two quantities:
- 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 you 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
Let's solve an interesting geometric problem. Suppose there is a triangular regular pyramid with some volume V 1. How many times should the size of this figure be reduced in order to obtain a pyramid homothetic to it with a volume three times smaller than the initial one?
Let's start solving the problem by writing the formula for the original regular pyramid:
V 1 = √3 / 12 * a 1 2 * h 1.
Let the volume of the figure, necessary according to the condition of the problem, be obtained if we multiply its parameters by the coefficient k. We have:
V 2 = √3 / 12 * k 2 * a 1 2 * k * h 1 = k 3 * V 1.
Since the ratio of the volumes of figures is known from the condition, we obtain the value of the coefficient k:
k = ∛ (V 2 / V 1) = ∛ (1/3) ≈ 0.693.
Note that we would get a similar value of the coefficient k for a pyramid of an arbitrary type, and not just for a regular triangular one.
Here we will analyze examples related to the concept of volume. To solve such tasks, it is imperative to know the formula for the volume of the pyramid:
S
h - height of the pyramid
The base can be any polygon. But in most problems on the exam, the condition, as a rule, is about the correct pyramids. Let me remind you of one of its properties:
The top of a regular pyramid is projected into the center of its base.
Look at the projection of the regular triangular, quadrangular and hexagonal pyramids (TOP VIEW):
You can read it on the blog, where you discussed the tasks associated with finding the volume of the pyramid.Consider the tasks:
27087. Find the volume of a regular triangular pyramid, the sides of the base of which are equal to 1, and the height is equal to the root of three.
S- the area of the base of the pyramid
h- pyramid height
Let's find the area of the base of the pyramid, this is a regular triangle. Let's use the formula - the area of a triangle is equal to half of the product of adjacent sides by the sine of the angle between them, which means:
Answer: 0.25
27088. Find the height of a regular triangular pyramid, the sides of the base of which are equal to 2, and the volume is equal to the root of three.
Concepts such as the height of the pyramid and the characteristics of its base are related by the volume formula:
S- the area of the base of the pyramid
h- pyramid height
We know the volume itself, we can find the area of the base, since we know the sides of the triangle, which is the base. Knowing the indicated values, we can easily find the height.
To find the area of the base, we will use the formula - the area of the triangle is equal to half the product of the adjacent sides by the sine of the angle between them, which means:
Thus, substituting these values into the volume formula, we can calculate the height of the pyramid:
The height is three.
Answer: 3
27109. In a regular quadrangular pyramid, the height is 6, the side edge is 10. Find its volume.
The volume of the pyramid is calculated by the formula:
S- the area of the base of the pyramid
h- pyramid height
We know the height. You need to find the area of the base. Let me remind you that the top of a regular pyramid is projected into the center of its base. The base of a regular quadrangular pyramid is a square. We can find its diagonal. Consider a right-angled triangle (highlighted in blue):
The segment connecting the center of the square with point B is the leg, which is half the diagonal of the square. This leg can be calculated by the Pythagorean theorem:
Hence BD = 16. Calculate the area of a square using the formula for the area of a quadrangle:
Hence:
Thus, the volume of the pyramid is equal to:
Answer: 256
27178. In a regular quadrangular pyramid, the height is 12, the volume is 200. Find the side edge of this pyramid.
The height of the pyramid and its volume and volume are known, so we can find the area of the square, which is the base. Knowing the area of a square, we can find its diagonal. Further, considering a right-angled triangle by the Pythagorean theorem, we calculate the lateral edge:
Find the area of the square (base of the pyramid):
Let's calculate the diagonal of the square. Since its area is 50, the side will be equal to the root of fifty and by the Pythagorean theorem:
Point O divides the diagonal BD in half, which means the leg of the right-angled triangle OB = 5.
Thus, we can calculate what the side edge of the pyramid is equal to:
Answer: 13
245353. Find the volume of the pyramid shown in the figure. Its base is a polygon, the adjacent sides of which are perpendicular, and one of the side edges is perpendicular to the base plane and is equal to 3.
As has been said many times - the volume of the pyramid is calculated by the formula:
S- the area of the base of the pyramid
h- pyramid height
The side edge perpendicular to the base is three, which means that the height of the pyramid is three. The base of the pyramid is a polygon with an area equal to:
Thus:
Answer: 27
27086. The base of the pyramid is a rectangle with sides 3 and 4. Its volume is 16. Find the height of this pyramid.
Back forward
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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 educate attention, accuracy, broaden the horizons of students.
Equipment and materials: computer, screen, projector, presentation “The volume of the pyramid”.
1. Frontal poll. Slides 2, 3
What is called a pyramid, the 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 to the points of the polygon.
This point called apex 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 face height the correct pyramid. The pyramid with at the bottom lies correct n-gon, a base height coincides with center of foundation called correct n-sided 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 remainder 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 top 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 SCBB pyramids 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 top of 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 the same 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 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. Exercise solution.
1) Task № 33 from the textbook by A.N. Pogorelova. Slides 7, 8, 9
On the side of the base? and the side edge b find the volume of the regular pyramid, at the base of which lies:
1) triangle,
2) a quadrangle,
3) hexagon.
In a regular pyramid, the height passes through the center of the circle around the base. Then: (Application)
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. 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 of 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 applied value.
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, the 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 the 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 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, the total number of phalanges of the fingers of each of which is also 14. This code also refers to the constellation Ursa Major, which includes our Sun, and in which it was once another star that destroyed Phaethon, 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 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 says that if you store jewelry under a pyramid, they self-clean and acquire a special shine, while 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, 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.
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