The top of the regular quadricutaneous pyramid. pyramid. Formulas and powers of the pyramid

Viznachennya. Bichna verge- This is a tricutnik, which has one coil lying at the top of the pyramid, and the side opposite to it meets the side of the base (richkutnik).

Viznachennya. Bechni ribs- these are the opposite sides of the side faces. The pyramid has as many ribs as there are as many cutas in the ornate bush.

Viznachennya. Height of the pyramid- This is perpendicular, descending from the top to the base of the pyramid.

Viznachennya. Apothem- This is perpendicular to the side face of the pyramid, descending from the top of the pyramid to the side of the base.

Viznachennya. Diagonal cut- through the pyramid with a plane that passes through the top of the pyramid and the diagonal of the base.

Viznachennya. Correct pyramid- this is a pyramid, in which the base is a regular rich body, and the height drops to the center of the base.

Volume and surface area of the pyramid

Formulas. Volume of the pyramid through the base area and height:

Power of the pyramid

Since all the side ribs are equal, a circle can be described around the base of the pyramid, and the center of the base is close to the center of the stake. The same perpendicular, descending from the top, passes through the center of the base (stake).

As the pork ribs are straight, all the smells are layered to the level of the base under the new kutas.

Beef ribs are equal, if they are created from the flatness of the base of the equal cut, or if you can describe a circle around the base of the pyramid.

If the side faces are extended to the base plane under one corner, then a column can be inscribed into the base of the pyramid, and the top of the pyramid is projected to its center.

As the side faces are piled up to the surface of the stand under one cut, then the apothems of the side faces are equal.

The power of the right pyramid

1. The top of the pyramid is evenly removed from all the corners of the base.

2. All side ribs are equal.

3. All side ribs are folded under the new cuts to the base.

4. Apothemes of all the great facets of the river.

5. The areas of all the edges of the plane.

6. All edges have new dihedral (flat) edges.

7. Around the pyramid you can describe a sphere. The center of the described sphere will be the point of the crossbar of the perpendiculars, which passes through the middle of the ribs.

8. You can inscribe a sphere before the pyramid. The center of the inscribed sphere will be the point of the crossbar of the bisectors that comes out from the corner between the edge and the base.

9. If the center of the inscribed sphere converges with the center of the described sphere, then the sum of flat parts at the top is equal to π or, for example, one part is equal to π/n, and n is the number of parts at the base of the pyramid.

Connection between the pyramid and the sphere

Around the pyramid, you can describe the sphere of the todi, if at the base of the pyramid there is a polyhedron, around which you can describe the circle (that sufficient brainpower is required). The center of the sphere will be the point of the crossbar of the planes, which runs perpendicularly through the middle of the side edges of the pyramid.

It is possible to describe a sphere using any kind of intricate or regular pyramid.

A sphere can be inscribed in a pyramid, since the bisectoral planes of the internal dihedral corners of the pyramid intersect at one point (sufficient brain space is required). This point will be the center of the sphere.

Link of a pyramid with a cone

A cone is called inscribed with a pyramid because their vertices meet, and the base of the cone is inscribed in the base of the pyramid.

The cone can be inscribed before the pyramid, since the pyramids are equal to each other.

The cone is called a description of the pyramid because their vertices meet, and the base of the cone is described of the base of the pyramid.

The cone can be described like a pyramid, since all the side ribs of the pyramid are equal to each other.

Link between the pyramid and the cylinder

A pyramid is called inscribed in a cylinder because the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.

A cylinder can be described around a pyramid, just as a circle can be described around the base of a pyramid.

Viznachennya. Truncated pyramid (pyramidal prism)- this is a rich face that is located between the base of the pyramid and the cross-cut plane parallel to the base. In this way, the pyramid has a larger base and a smaller base, which is similar to the larger one. The side faces are trapezoidal.

Viznachennya. Tricutaneous pyramid- this is a pyramid with three sides and a base with several tricubitules.

There are four sides, four vertices and six edges, so even if the two edges do not touch each other, they do not stick together.

The skin apex consists of three edges and ribs that create triangular cut.

The cut that connects the vertex of the trihedron with the center of the protagonal face is called median(GM).

Bimedianoy called a cut that connects the middle of the prostrate ribs that do not stick together (KL).

All bimedians and medians of the tiger face move at the same point (S). In this case, the bimedians are divided in half, and the medians are 3:1, starting from the top.

Viznachennya. Pokhila pyramid- this pyramid in one of the ribs creates an obtuse cut (β) at the base.

Viznachennya. Rectcutaneous pyramid- this is a pyramid in which one of the side faces is perpendicular to the base.

Viznachennya. Gostrokutna pyramid- this is a pyramid in which the apothem is more than half of the side of the base.

Viznachennya. Dull pyramid- this is a pyramid in which the apothem is less than half the side of the base.

Viznachennya. Regular tetrahedron- a sided player, whose all sides are equal-sided. He is one of the five correct rich people. In a regular tetrahedron, all dihedral edges (between the faces) and trihedral edges (at the vertex) are equal.

Viznachennya. Straight tetrahedron It is called a trihedron where there is a straight cut between three ribs at the apex (the ribs are perpendicular). Three faces are created straight-cut three-cut And the edges have rectilinear tricubitules, and the base has a long tricubitule. The apothem be between the ancient half of the side of the base, where the apothem falls.

Viznachennya. Even-sided tetrahedron It is called a triangular, whose side faces are equal to each other, and the base is a regular tricubitine. Such a tetrahedron has edges at the equifemoral tricutules.

Viznachennya. Orthocentric tetrahedron It is called a trihedron, in which all the heights (perpendiculars) that extend from the top to the prolongation edge intersect at one point.

Viznachennya. Zirkov's pyramid It is called a richhedron, which has a star as its basis.

Viznachennya. Bipyramide- a polyhedron, which consists of two different pyramids (pyramids can also be cut), which form the underlying base, and the tops lie along the different sides of the base plane.

How do we call it a pyramid? First of all, it is a rich-hedron. In another way, at the base of this rich face there is a large rich side, and the sides of the pyramid (the side faces) obliquely form the shape of the tricuts, which converge in one joint vertex. Axis Now, having understood the term, it is clear how to find the surface area of the pyramid.

It is clear that the surface area of such a geometric body is the sum of the area of the base and all of its side surface.

Calculation of the area of the base of the pyramid

The choice of the rosemary formula lies in the form of the rich-cuttle, which lies at the basis of our pyramid. It may be correct, but it may be the same, or it may be incorrect. Let's take a look at the options.

The basis is the correct richness

From the school course you can see:

the area of the square on the other side of the square;
The area of an even-sided tricube is equal to the square of its side divided by 4 and multiplied by the square root of three.

There is a simple and straightforward formula for dividing the area of any regular septum (Sn): you need to multiply the value of the perimeter of that sac (P) by the radius inscribed in the new stake (r), and then divide the result by two: Sn = 1/2P* r.

It is based on an irregular richness

The scheme for finding its area lies in the fact that the entire skin is divided into tricutules, calculating the area of the skin from them using the formula: 1/2a * h (where a is the base of the tricutule, h is the height lowered onto the base), adding up all the results.

Area of the side surface of the pyramid

Now we will open up the area of the side surface of the pyramid, then. the sum of the area of the ears and the side sides. There are also 2 options here.

Let us have a sufficient pyramid, then. so, on the basis of this - an incorrect rich-cutnik. Then calculate the surface area of the skin and summarize the results. Since the side sides of the pyramid can only be tricutaneous, then the structure will follow the formula: S = 1/2a * h.
May our pyramid be correct, then. At its base lies a regular rich body, and the projection of the top of the pyramid appears at its center. To calculate the flat side surface (Sb), it is enough to know half of the perimeter of the base (P) per height (h) of the side (the same for all faces): Sb = 1/2 P * h. The perimeter of the orchard is determined by the addition of all its sides.

The entire surface area of a regular pyramid will be found by summing up the surface area of the base with the area of the entire side surface.

Apply it

For example, the algebraic area of the surface of several pyramids is computable.

Surface area of the tricutaneous pyramid

The basis of such a pyramid is the tricubitus. Using the formula So = 1/2a*h we know the area of the base. This formula is used to find the flat skin edge of the pyramid, also has a tricutaneous shape, and there are 3 areas: S1, S2 and S3. The area of the side surface of the pyramid is the sum of all areas: Sb = S1 + S2 + S3. Having flattened the sides and the base, we extract the surface area of the pyramid: Sp = So + Sb.

Surface area of the quadricutaneous pyramid

The area of the barrel surface is the sum of 4 additions: Sb = S1 + S2 + S3 + S4, the skin of which is calculated according to the formula of the area of the tricutaneous area. And the area of the base happens to be shukati, depending on the shape of the chotirikutnik - right and wrong. The total surface area of the pyramid is again shown by the folding plane of the base and the total surface area of the given pyramid.

The concept of a pyramid has been studied by scientists long before geometry has been developed. This is due to the famous great Egyptian wonders of the world. Therefore, beginning to become accustomed to this miraculous richness, most scientists are already clearly manifesting themselves. All fortune telling reminders are in the correct shape. So what? correct pyramid, and what kind of power there may be and will be far away.

In contact with

Viznachennya

The value of the pyramid can be adjusted. Starting from ancient times, there is little popularity.

For example, Euclid defined it as a bodily figure that is made up of planes that begin to converge at a single point.

Heron gave a more precise formulation. Vin insisted on what to do, like forms the basis and flatness of the tricutaneous appearance, what to converge on one point.

Spiraling in the current darkness, the pyramid is represented as a spacious rich-hedron, which consists of a singing k-cutnik and k flat figures of a tricutaneous shape, which has one hidden point.

Let's look into the report, What are the elements that make up the following:

k-kutnik is considered the basis of the figure;
figures of a 3-piece form protrude as the edges of the barrel part;
the upper part, from which the cob of the bean element is taken, is called the apex;
all the cuts that connect the top are called ribs;
If from the top to the surface of the figure you lower it directly below 90 degrees, then this part is placed in the inner space - the height of the pyramid;
in any element you can draw a perpendicular to our polyhedron, called an apothem.

The number of edges is calculated using the formula 2*k where k is the number of sides of the k-kutnik. How many faces of such a rich face, like a pyramid, can be calculated using the additional expression k+1.

Important! A pyramid of regular shape is a stereometric figure, the base of which is a k-column with equal sides.

Main authorities

Correct pyramid there is no one in power, whoever is in power is less powerful than her. We override it:

The basis is a figure of correct shape.
The edges of the pyramid, which intersect the side elements, have equal numerical values.
The bicuspid elements are the iso-femoral tricutules.
The base of the height of the figure goes to the center of the rich body, in which it is simultaneously the central point, inscribed and described.
All side ribs are stretched to the base plane under a new cut.
All natural surfaces are washed down to the base.

Once again, all the over-insurance authorities will have a lot of trouble with the calculation of elements. Following the instructions of the authorities, we are deprived of respect for two signs:

In this case, if the rich cutlet fits into the circle, the side faces meet the base of the cut.
When describing the stake of the rich man, all the edges of the pyramid that come out from the top are equal to the bottom and equal to the base.

The basis is a square

The most accurate pyramid is correct - a polyhedron, which has a square as its base.

She has many natural facets, which are equal in appearance.

On the surface, a square is depicted, but it is grounded on all the authorities of the correct chotirikutnik.

For example, if you need to relate the side of a square to its diagonal, then you can formulate the following formula: the diagonal is equal to the side of the square by the square root of two.

The basis is the correct trikutnik

A regular tricutaneous pyramid is a polyhedron, which is based on a regular tricubitine.

Since the base is a regular tricut, and the side ribs are equal to the ribs of the base, then such a figure is called a tetrahedron.

All faces of the tetrahedron have equilateral 3-pieces. In this situation, it is necessary to know the timing and not waste an hour on them when calculating:

cut the edges to the base of the old 60 degrees;
the size of all internal edges is also set to 60 degrees;
if the edge could become the basis;
, drawn in the middle of the figure, the same elements

Transection of the richhedron

Any rich-hedron will be separated number of views per section flat Most often, school geometry courses are taught in two ways:

osov;
parallel to the base.

The axial cross-section is trimmed when crossing with the flat of the rich-hedron, which passes through the top, side ribs and all. Once the whole height is drawn from the top. The upper surface is bordered by webbing lines and edges, resulting in a removable tricut.

Respect! The correct pyramid has an axial strut and an isosfemoral tricumus.

If the cross-section runs parallel to the base, then as a result another option is chosen. This type has a figure that is similar to the core.

For example, if the base is a square, then the crosspiece parallel to the base will also be a square, only of smaller dimensions.

With a strong command behind such minds, there are signs of the power of similar figures, based on Thales' theorem. We first need to determine the coefficient of similarity.

If the plane is drawn parallel to the base and meets the upper part of the richhedron, then a correct truncated pyramid is cut out of the lower part. It would seem that the bases of the rich-hedron are similar rich-edges. And here the side faces are equal-sided trapezoids. The axial retina is also equilateral.

In order to calculate the height of the cut richhedron, it is necessary to draw the height in the axial section, or in the trapezium.

Surface area

The main geometric tasks that must be covered in a school geometry course are finding the surface area and surface area of the pyramid.

Surface area values are divided into two types:

flatness of the barrel elements;
flatter than all surfaces.

From the name itself it’s clear what’s going on. The barrel surface includes the barrel elements. This means that to find it, it is necessary to simply fold the flat surfaces of the hips, so that the flat surfaces of the isosfemoral tricuputae are flattened. Let's try to derive the formula for the surface area of the elements:

The area of the isosfemoral tricuput is Str=1/2(aL), where a is the side of the base, L is the apothem.
A large number of barrel flats lie in the form of the k-th braid at the base. For example, a regular pyramid extends over almost every surface. So, it is necessary to fold four figures flat Sside=1/2(aL)+1/2(aL)+1/2(aL)+1/2(aL)=1/2*4a*L. Viraz is simple with this method, the value 4a = Rosn, de Rosn - the perimeter of the base. And viraz 1/2*Rosn є її around the perimeter.
Therefore, let’s remember that the area of the side elements of the regular pyramid is around the perimeter of the base of the apothem: Side = Rosn * L.

The area of the total surface of the pyramid is the sum of the areas of the barrel surfaces and the base: Sp.p. = Sside + Smain.

Since there are flat bases, then the formula seems to be similar to the type of rich cutter.

Volume of the correct pyramid The ancient addition of a flat surface to a height divided into three: V = 1/3 * Sbas * H, de H is the height of the rich face.

What is the correct pyramid in geometry?

The power of the correct four-way pyramid

The total area of the side surface of the pyramid is the sum of the area of its side faces.

The tricutaneous pyramid has two types of edges - the tricutaneous base and the tricutaneous ones with a lateral apex, which create a side surface.
To make the cob, you need to open up the area of the sides. For this purpose, you can use the formula for the surface area of the tricutaneous pyramid, and you can also quickly use the formula for the surface area of the tricutaneous pyramid (only at the same time, if the tricucutineum is regular). If the pyramid is regular and it shows the extension of the edge a of the base and drawn to the next apothem h, then:

Since the dovzhina of the edge from the correct pyramid and the dovzhina of the side of the base are given behind the heads, you can find out the values behind the offensive formula:

If you have given the length of the rib at the base and the sharp cut to lie at the top, then you can expand the area of the butt surface along the lines of the square of side a to the double cosine of the half cut α:

Let's take a look at the expansion of the flat surface of the near-cut pyramid through the side rib and side of the base.

Zavdannya: let the correct pyramid be given. The length of the edge b = 7 cm, the length of the side of the base a = 4 cm. Let us substitute the given value with the formula:

We showed the dimensions of the area of one side of the barrel for the correct pyramid. Apparently. To find the area of the entire surface, you need to multiply the result by the number of faces, that is, by 4. If the pyramid is sufficient and its faces are not equal to each other, then it is necessary to expand the area for the skin edged sides. Since it is based on a rectilinear or parallelogram, it is possible to guess their power. The sides of these figures are parallel in pairs, and the sides of the pyramid will also be the same in pairs.
The formula for the flat base of the four-cut pyramid must be based on what kind of cut-throat is at the base. If the pyramid is correct, then the area of the base is covered by the formula, since the basis is a rhombus, you need to guess how it is found. Since the base is a straight cutter, then it will be easy to find its area. It is enough to know more than one side of the basics. Let's take a look at the layout of the plane of the base of the almost identical pyramid.

Commandment: Find a pyramid, at the base of which lies a rectum with sides a = 3 cm, b = 5 cm. An apothem is lowered to the skin side from the top of the pyramid. h-a = 4 cm, h-b = 6 cm. The top of the pyramid lies on the same line as the cross point of the diagonals. Find the area of the pyramid again.
The formula of a flat pyramid is composed of the sum of the flat faces and the flat base. For starters, I know the basics:

Now let's look at the edges of the pyramid. They smell in pairs, however, so the height of the pyramid exceeds the point of the crossbar of the diagonals. So, in our pyramid there are two triangles with a base a and height h-a, as well as two triangles with a base b and height h-b. Now we know the area of the tricubitus using the familiar formula:

Now let's finish the butt out of the structure of the flattened pyramid. For our pyramid with Orthocutaneum at the base, the formula will look like this:

Before learning about this geometric figure and its power, we need to talk about it in different terms. Whenever a person hears about the pyramid, they see the grandeur of sporudi in Egypt. This is how the simplest of them look. There are different types and shapes, which means the calculation formula for geometric shapes will be different.

Pyramid is a geometric figure What does a number of faces mean? In essence, it is the same tricutaneous structure, which is based on a tricutaneous structure, and on the sides there are expanded tricutaneous structures that unite at one point - the apex. The figure comes in two main types:

correct;
truncated.

In the first case, the basis is the correct richness. Here all the natural surfaces are level between themselves and themselves will please the eye of a perfectionist.

In another type, there are two bases - a large one at the very bottom and a small one between the top, which repeats the shape of the main one. In other words, there is a truncated pyramid with a rich side with a crossbar, created parallel to the base.

Terms and meaning

Basic terms:

Correct (even-sided) jersey- a figure with three equal cuts and equal sides. And here everything is hovering at 60 degrees. The figure is the simplest of regular rich sides. Since this figure lies at the base, then such a polyhedron is called a regular tricutaneous. Since the base is a square, the pyramid is called a regular square pyramid.
Vertex- The best point is where the edges converge. The height of the vertex is determined by a straight line that extends from the vertex to the base of the pyramid.
Edge- One of the areas of the rich-combustion area. You can have the appearance of a tricubitus for a truncated pyramid or a trapezoid for a truncated pyramid.
Pererez- A flat figure that is created as a result of dissection. Do not get confused with the cut, because the cut shows those who are behind the webbing.
Apothem- A section from the top of the pyramid to the base. It is also the height of the same boundary, where the other point of height is located. Given the designation is fair without a hundred-so-right polyhedron. For example, if the pyramid is not truncated, then the edge will be a three-piece. At this point, the height of this trikutnik will become an apothem.

Formulas are flat

Find the area of the side surface of the pyramid Any type can be done in any number of ways. Since the figure is not symmetrical and has a large body with different sides, then in this case it is easier to calculate the ground area of the surface through the totality of the surfaces. In other words, you need to rub the area of the skin and fold them at the same time.

Also, whatever parameters are visible, there may be necessary formulas for calculating a square, trapezoid, square, etc. The formulas themselves have different variations It also means importance.

With the right figure, the area is much simpler. It is enough to know just a few key parameters. In most cases, the required calculations are the same for such figures. Subsequent formulas will then be introduced. Otherwise, I would have had to write everything down on so many sides that it would only get lost and messy with the panel.

Basic formula for calculation The flat surface of the regular pyramid will look like this:

S=½ Pa (P is the perimeter of the base, and is the apothem)

Let's take a look at one of the butts. The bagatihedron forms a base with sections A1, A2, A3, A4, A5, and all the dimensions are 10 cm. The apotheme is 5 cm. For the cob, you need to know the perimeter. So since all five faces of the base are the same, it can be calculated as follows: P = 5 * 10 = 50 cm. The following is the basic formula: S = ½ * 50 * 5 = 125 cm squared.

Area of the side surface of the regular tricutaneous pyramid calculate as easily as possible. The formula looks like this:

S =½* ab *3, where a is apothem, b is between the bases. The multiply of three here means the number of edges of the base, and the first part means the area of the side surface. Let's take a look at the butt. Given a figure with an apothem of 5 cm and a base edge of 8 cm. Calculated: S = 1/2 * 5 * 8 * 3 = 60 cm for a square.

Area of the barrel surface of a truncated pyramid It’s easier to count the little things. The formula looks like this: S =1/2*(p_01+ p_02)*a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Let's take a look at the butt. It is permissible that for a slightly cut figure the dimensions of the sides of the bases are 3 and 6 cm, the apotheme is 4 cm.

Here, for the cob, calculate the perimeter of the bases: p_01 = 3 * 4 = 12 cm; р_02=6*4=24 div. It is no longer necessary to introduce the main formula and can be removed: S = 1/2*(12+24)*4=0.5*36*4=72 div per square.

In this way, you can find out the area of the barrel surface of a regular pyramid of any kind. Be respectful and don’t get confused This is calculated from the full area of the entire rich face. But if you still need to work it out, it is enough to calculate the area of the largest base of the rich face and add it to the flat side surface of the rich face.