For solutions similar tasks You definitely need to know the formula for the volume of a pyramid:
S
h– height of the pyramid
The base can be any polygon. But in most problems Unified State Exam speech the condition, as a rule, refers to regular pyramids. Let me remind you of one of its properties:
Vertex regular pyramid projected to the center of its base
Look at the projection of a regular triangular, quadrangular and hexagonal pyramid(VIEW FROM ABOVE):
You can on the blog, where problems related to finding the volume of a pyramid were discussed.
Let's consider the tasks:
27087. Find the volume of the correct triangular pyramid, the sides of which are equal to 1, and whose height is equal to the root of three.
S– area of the base of the pyramid
h– height of the pyramid
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 the product of adjacent sides and the sine of the angle between them, which means:
Answer: 0.25
27088. Find the height of a regular triangular pyramid whose base sides are equal to 2 and whose volume is equal to the root out of three.
Concepts such as the height of a pyramid and the characteristics of its base are related by the volume formula:
S– area of the base of the pyramid
h– height of the pyramid
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 use the formula - the area of a triangle is equal to half the product of adjacent sides and the sine of the angle between them, which means:
Thus, by 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, side rib equals 10. Find its volume.
The volume of the pyramid is calculated by the formula:
S– area of the base of the pyramid
h– height of the pyramid
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 triangle (highlighted in blue):
The segment connecting the center of the square with point B is the leg, which equal to half diagonals of a square. We can calculate this leg using the Pythagorean theorem:
This means BD = 16. Let’s calculate the area of the square using the formula for the area of a quadrilateral:
Hence:
Thus, the volume of the pyramid is:
Answer: 256
27178. In a regular quadrangular pyramid, the height is 12 and the volume is 200. Find the side edge of this pyramid.
The height of the pyramid and its volume are known, which means we can find the area of the square, which is the base. Knowing the area of a square, we can find its diagonal. Next, considering a right triangle using the Pythagorean theorem, we calculate the side edge:
Let's 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 according to the Pythagorean theorem:
Point O divides diagonal BD in half, which means leg right 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 plane of the base and equal to 3.
h- height of the pyramid
S- base area ABCDE
V- volume of the pyramid
In geometry, a pyramid is a body that has a polygon at its base, and all its faces are triangles with a common vertex. Depending on which figure lies at the base, pyramids are divided into triangular, quadrangular, pentagonal, etc. In addition, there are regular, truncated, rectangular and arbitrary pyramids. Formula for calculating volume this body is not complex and is known to everyone from school course geometry.
A classic example of the use of pyramids in architecture is egyptian tombs pharaohs, many of whom have exactly this shape. It should be noted that similar structures (albeit somewhat modified) are found in other parts of the world and countries, for example, in Mexico and China, and it is characteristic that almost everywhere they are either tombs or religious buildings. Of course, when designing them, the ancient architects hardly sought to determine the volume of their creations, but their “followers” certainly had to do this.
Modern architects also sometimes create pyramidal buildings, in which social and cultural facilities are most often located (shopping and entertainment complexes, exhibition galleries etc.), and at the same time it is necessary to calculate the volume of these structures so that they comply with accepted building codes, rules and regulations. Besides, exact value This value is required in order to most rationally place utility lines in the building.
IN last years Greenhouses with pyramid shape. Most often, they are built from transparent polycarbonate and, according to their developers, have significant advantages over traditional ones. Since for the same total area base, the volume of air contained in them is approximately three times less, and it heats up significantly faster. In addition, it is distributed more rationally, since there is also less space for the warmest gas accumulating at the top in a pyramidal greenhouse.
Pyramids can often be found in ordinary apartments, country houses and cottages. The bells of kitchen hoods, which are used to effectively remove hot air, smoke and fumes from rooms, often have their shape. Those elements of ventilation systems that are used to connect air ducts of different cross-sections are often made in the form of truncated pyramids.
One of the most popular puzzles is the so-called “ Meffert pyramid", which is often called " Rubik's tetrahedron", although the Hungarian architect and inventor has nothing to do with it. Each of its faces is divided into nine multi-colored regular triangles, and the player’s goal is to bring the toy into such a form that on each individual face all its elements have the same color.
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.
However, let's try to give her 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 differently - as geometric figure, having a polygon at its base, and triangles united by a common vertex as its side faces.
Since the pyramid is spatial figure, then she has one too quantitative characteristic, as volume. The volume of the pyramid is calculated using well well-known formula volume equal to a 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 from all the pyramids are the correct ones, which have at their base regular polygon. As for , it should “end” in the center of the base.
When irregular polygon in 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 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 quadrangular pyramid, if it is regular, square the length of the side of a regular quadrangle (square) at the base and, multiplying by the height of the pyramid, divide the resulting product 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 the truncated pyramids obtained by cross-section of the pyramid parallel to the base flat. 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, equal to one sixth of the module mixed product given vectors.