The entire surface of the irregular pyramid. Pyramid area

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Pyramid- one of the varieties of the algae, made of algae and tricutaneous, which lie at the base and its faces.

Moreover, at the top of the pyramid (at one point) all the edges meet.

In order to calculate the area of the pyramid, you must first determine that its surface is made up of a number of tricuts. And their areas can be easily known, stastosovuchi

different formulas. It is important to know what data we know from the trikutniki, we understand their area.

Let's reconsider the following formulas, using which you can find the area of tricuticles:

S = (a * h) / 2 . At whose height we see the height of the tricutaneous h , yaka lowered on the side a .
S = a*b*sinβ . Here are the sides of the tricut a , b , And where is it between them? β .
S = (r * (a + b + c)) / 2 . Here are the sides of the tricut a, b, c . Radius of the stake, as it is inscribed in the tricut - r .
S = (a * b * c) / 4 * R . Radius, the described stake near the tricutaneous. R .
S = (a*b)/2 = r² + 2*r*R . This formula should be consolidated only if the tricut is straight.
S = (a²*√3)/4 . This formula is condensed to an even-sided trikutnik.

Moreover, since the area of all the tricuputs, which are the edges of our pyramid, is expanded, we can calculate the area of its surface. For this purpose we will go through a list of more formulas.

In order to calculate the area of the side surface of the pyramid, there is no question of using any folding device: it is necessary to find out the sum of the area of all the tricupus. Virazimo tse formula:

Sp = ΣSi

Here Si є flatter than the first tricut, and S P - The area of the side surface of the pyramid.

Let's take a look at the butt. A correct pyramid is given, its side faces are created by a number of equilateral tricupus,

« Geometry is the most powerful way to enhance our mental abilities».

Galileo Galilei.

and the square is the basis of the pyramid. Moreover, the edge of the pyramid extends 17 cm. We know the area of the side surface of this pyramid.

It is sized as follows: it is clear that the edges of the pyramid are tricutaneous, the stench is equilateral. We also know how long the edge of this pyramid is. Let's go out, where the trikutniks are waving on their equal sides, their day is 17 days.

To calculate the area of the skin and the tricutubes, you can use the following formula:

S = (17² * √ 3) / 4 = (289 * 1.732) / 4 = 125.137 cm²

Since we know that the square lies at the base of the pyramid, it turns out that we may have equal-sided tricutules. And this means that the area of the barrel surface of the pyramid is easy to calculate using the following formula: 125.137 cm² * 4 = 500.548 cm²

Our answer is now: 500.548 cm² - this is the area of the barrel surface of this 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 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 out, you need to calculate the area of the largest base of the rich-hedron and add it to the flat side surface of the rich-hedron.

Video

This video will help you find out the area of the side surface of different pyramids.

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Instructions

First of all, please understand that the outer surface of the pyramid is represented by decal tricutaneous elements, which can be found using various formulas, including the following data:

S = (a * h) / 2 de h - height, lowered to side a;

S = a*b*sinβ, where a, b are the sides of the tricut, a is the cut between these sides;

S = (r * (a + b + c)) / 2, where a, b, c are the sides of the tricut, and r is the radius of the stake inscribed in this tricut;

S = (a*b*c)/4*R, where R is the radius of the tricutaneous tree described around the stake;

S = (a * b) / 2 = r + 2 * r * R (as trikutnik is straight-cut);

S = S = (a²*√3)/4 (as the tricut is even-sided).

In fact, there are no basic formulas for finding the tricuput area.

Having studied the area of all tricuputs, which is the edges of the pyramid, using additional meanings of formulas, you can proceed to calculating the area of the pyramid. It’s quite simple to do: you just need to fold the surfaces of all the three-pieces to create a solid surface of the pyramid. The formula can be expressed as follows:

Sp = ΣSi, where Sp is the area of the barrel, Si is the area of the i-th tricube, which is part of the barrel surface.

For greater clarity, you can look at a small butt: there is a regular pyramid, the side faces are created by equal-sided tricubitules, and at the base there is a square. The length of the edge of this pyramid is 17 cm. It is necessary to find the area of the side surface of this pyramid.

Solution: from the bottom of the edge of this pyramid, it is clear that their boundaries are equal-sided tricubitules. In this way, we can say that all sides of all the tricubitules on the butt surface are 17 cm. Therefore, in order to expand the area of each of these trikulets, it is necessary to formulate the formula:

S = (17² * √ 3) / 4 = (289 * 1.732) / 4 = 125.137 cm²

It appears that the base of the pyramid is a square. In this manner, it is clear that these equal-sided trikutniks were chosen. Then the area of the side surface of the pyramid is covered as follows:

125.137 cm² * 4 = 500.548 cm²

Conclusion: the area of the barrel surface of the pyramid becomes 500.548 cm²

The area of the side surface of the pyramid is calculable. Under the barrel surface there is a noticeable amount of square edges of the barrel. If you are on the right with a regular pyramid (one that is based on a regular ornate structure, and the apex is projected into the center of this ornate structure), then to calculate the entire side surface it is sufficient to multiply the perimeter of the base (then the sum will be equal to Their sides of the rich body that lies at the base of the pyramid ) by the height of the barrel face (otherwise called the apothem) and divide the resulting value by 2: Sb = 1/2P*h, where Sb is the area of the barrel surface, P is the perimeter of the base, h is the height of the barrel face (apothem).

If you have a large pyramid in front of you, you will have to carefully calculate the areas of all the faces and add them up. The fragments with the side faces of the pyramid are the tricumulus, use the formula for the area of the tricumulus: S = 1/2b * h, where b is the base of the tricumulus, and h is the height. If all the facets are flat, the flatness of them is removed in order to reduce the area of the side surface of the pyramid.

Then it is necessary to calculate the area of the base of the pyramid. The choice of formula for development depends on which richness lies at the base of the pyramid: correct (the same, all sides of which may be the same) and incorrect. The area of a regular orchardcut can be calculated by multiplying the perimeter by the radius of the stake inscribed in the richcut and dividing the value by 2: Sn=1/2P*r, where Sn is the area of the richcut, P is the perimeter, and r is the radius of the stake inscribed in the richcut. .

A truncated pyramid is a polyhedron, which is created by a pyramid and a crossbar parallel to the base. Finding the area of the side surface of the pyramid is very easy. It’s even simpler: the area of the old dobutku is half the sum of the bases. Let's take a look at the butt from the flat surface of the barrel. Let's say the correct pyramid is given. Towards the end of the base, b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of the side surface of the pyramid, you first need to know the perimeter of the bases. For the great base, the area is more advanced: p1=4b=4*5=20 cm. For the smaller base, the formula will be the same: p2=4c=4*3=12 cm. Then, the area is more advanced: s=1/2(20+12)* 4=32/2*4=64 div.