How to know the surface of the pyramid formula. The square of the Chotirikutnoi Pyramid. The surface area of the tricot pyramid

Appointment 1. The pyramid is called correct, as if it were the basis, the correct bagatokutnik, with which the top of such a pyramid is projected to the center of the foundation.

Appointment 2. The pyramid is called correct, because its base is a regular bagatokutnik, and the height passes through the center of the base.

Elements of the right pyramid

height apothem. The little one is marked as a vіdrіzok ON
Krapka, which hits the ribs and does not lie at the base area, is called top of the pyramid(O)
Trikutniks, who knit the opposite side with the base and one of the vertices, which run with the apex, are called bіchnymi faces(AOD, DOC, COB, AOB)
Vіdrіzok perpendicular drawn through the top of the pyramid to the plane of the її base curls of the pyramid(OK)
Diagonal cross section of the pyramid- tse peretin, scho to pass through the top and the diagonal of the base (AOC, BOD)
Bagatokutnik, who cannot lay the top of the pyramid, is called the basis of the pyramid(ABCD)

Yakshcho on a stand correct pyramids lie trikutnik, chotirikutnik thinly. then she is called correct knitted , which and etc.

Trikutna pyramid є chotihedral - tetrahedron.

The power of the right pyramid

For the accomplishment of the task, it is necessary to know the power of the okremikh elements, as in the minds of the sound fall down, to that which is respected, that the student is guilty of the nobility of the head.

bichni ribs are equal among themselves
apothemies equal
bіchnі verge rіvnі between themselves (with tsoma, vіdpovіdno, equal їх areas, bіchnі sides of that foundation), then stinks є equal trikutniks
all bіchnі faces є equal equal-femoral tricots
if you can write the right pyramid, so describe the sphere
as the centers of the inscribed and described spheres zbіgayutsya, then the sum of flat kutіv at the top of the pyramid is more expensive π, and the skin of them is clearly π / n, de n - the number of sides of the base
the square of the square on the surface of the regular pyramid and the half of the perimeter of the base on the apotheme
for the basis of the correct pyramid, you can describe the stake (div. also the radius of the described stake of the tricot)
all the bіchnі faces utvoryuyut іz the flat foundation of the correct pіramіdі іvnі kuti.
all the heights of the bіchnyh faces are equal among themselves

Vkazіvki until the end of the day. Authority, guidance, responsibility to help in a practical solution. If it is necessary to know the cuti of the cheeky faces, their surface, etc., then the ignominious technique is to break up all the volumes of figures on the okremі flat figures and the stagnation of their powers for the recognition of the okremіh elementsіvіrіmіdі, oskіlki for fіlіkіlі elements richly

It is necessary to break the whole volume of the figure on the edge of the elements - tricots, squares, ribs. Dali, up to the last few elements, zastosuvat knowledge of the course of planimetry, which will significantly simplify the knowledge of vіdpovіdі.

Formulas for the correct pyramid

Formulas for knowing the volume and surface area:

Appointment:
V - obsyag piramidi
S - base area
h - pyramid height
Sb - square surface
a - apothem (do not stray from α)
P - base perimeter
n - number of sides of the base
b - dozhina of the rib
α - flat kut at the top of the pyramid

Tsya formula of knowledge is obligatory only for correct pyramids:

, de

V - obsyag correct ї piramidi
h - height of the correct pyramid
n - the number of sides of the correct bagatokutnik, which is the basis of the correct pyramid
a - dozhina of the side of the correct bagatokutnik

Correctly truncated pyramid

If it is necessary to carry out a resection, a parallel support of the pyramid, then the body, laid between them by the tabular surface, is called truncated pyramid. Tsej pererіz for truncated pyramid є one of її pіdstav.

The height of the bіchnoї verge (like є rіvnobokoy trapezієyu), called - apothem of the right truncated pyramid.

A shortened pyramid is called correct, like a pyramid, for which it was taken away, it is correct.

Stand between the foundations of a truncated pyramid called the height of the truncated pyramid
mustache facets of a regular truncated pyramidє equilateral (equivalent) trapezium

Notes

Div. also: okremi vipadki (formulas) for the correct pyramid:

How to speed up the theoretical materials introduced here to complete your task:

Appointment of the pyramid

pyramid- tse bagatohedron, the basis of which is a bagatokutnik, and the edges of yoga are trikutniks.

Online calculator

Varto zupinitisya on the designated warehouse pyramids.

She, yak and in other bagatohedrons, є ribs. The stench converge to one point, as it is called pinnacle pyramids. On the її pedestal, there may be a large bagatokutnik. edge It is called a geometric figure, adorned with one of the sides of the foundation and the two nearest ribs. Our vipad has a tsetrikutnik. high The pyramids are called in the middle of the plane, in which lie the foundation, up to the top of the bagatohedron. For the correct pyramid, understand apothemies- Tse perpendicular, omissions from the top of the pyramid to the її base.

See the pyramid

Іsnuyut 3 types of pyramids:

Rectangular- the one that has a rib that makes a straight cut with the base.
correct- it has a base - a correct geometric position, and the top of the bagatokutnik itself is a projection to the center of the base.
tetrahedron- Pyramid, stacked with tricoutniks. Moreover, leather from them can be taken as a basis.

Surface area formula of the pyramid

For knowing the total area of the surface of the pyramid, it is necessary to fold the area of the base surface and the area of the base.

Let's just make the right step in the right pyramid, so let's take care of it. Let's calculate the total area of the surface of such a pyramid. The square of the white surface of the door:

S bіk = 1 2 ⋅ l ⋅ p S_(\text(side))=\frac(1)(2)\cdot l\cdot pS bik = 2 1 ⋅ l ⋅p

l l l- apothem of the pyramid;
pp p- the perimeter of the base of the pyramid.

Total surface area of the pyramid:

S = S bіk + S main S = S_ (text (side)) + S_ (text (main))S=S bik + S main

S bіk S_(text(side)) S bik - area of the bіchnoi surface of the pyramid;
S main S_(text(main)) S main - area of the base of the pyramid.

An example of solving problems.

Butt

To know exactly the area of the tricot pyramid, as if the apothem is dear 8 (div.), and to lie the equal-sided trikutnik on the side 3 (div.)

Solution

L = 8 l = 8 l =8
a=3 a=3 a =3

We know the perimeter of the base. Shards are based on an equal-sided tricoutnik on the side a a a, then yogo perimeter pp p(Sum of all sides):

P = a + a + a = 3 ⋅ a = 3 ⋅ 3 = 9 p=a+a+a=3\cdot a=3\cdot 3=9p=a +a +a =3 ⋅ a =3 ⋅ 3 = 9

Todi bichna of the pyramid square:

S bіk = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 8 ⋅ 9 = 36 S_(\text(side))=\frac(1)(2)\cdot l\cdot p=\frac(1)(2) \cdot 8\cdot 9=36S bik = 2 1 ⋅ l ⋅p=2 1 ⋅ 8 ⋅ 9 = 3 6 (Div. sq.)

Now we know the area of the foundations of the pyramid, so I am the area of the trikutnik. At our point of view, the equal-sided tricot can be calculated using the formula:

S main = 3 ⋅ a 2 4 S_(\text(main))=\frac(\sqrt(3)\cdot a^2)(4)S main = 4 3 ⋅ a 2

A a a- the side of the tricot.

We take:

S main = 3 ⋅ a 2 4 = 3 ⋅ 3 2 4 ≈ 3.9 S_(\text(main))=\frac(\sqrt(3)\cdot a^2)(4)=\frac(\sqrt(3 ) )\cdot 3^2)(4)\approx3.9S main = 4 3 ⋅ a 2 = 4 3 ⋅ 3 2 ≈ 3 . 9 (Div. sq.)

Povna area:

S = S bіk + S main ≈ 36 + 3.9 = 39.9 S=S_(text(side))+S_(text(main))\approx36+3.9=39.9S=S bik + S main ≈ 3 6 + 3 . 9 = 3 9 . 9 (Div. sq.)

Suggestion: 39.9 cm. sq.

Another butt, troch folded.

Butt

Supporting the pyramid is a square of area 36 (div. sq.). The apothem of the bagatohedron is 3 times larger for the side of the base a a a. Know exactly the surface area of the figure.

Solution

S quad = 36 S_(text(quad))=36S quad = 3 6
l = 3 ⋅ a l=3\cdot a l =3 ⋅ a

We know the base of the foundation, that is the base of the square. Yogo area that dozhina side pov'yazanі:

S quad = a 2 S_(\text(quad))=a^2S quad = a 2
36 = a2 36 = a^2 3 6 = a 2
a=6 a=6 a =6

We know the perimeter of the base of the pyramid (that is the perimeter of the square):

P = a + a + a + a = 4 ⋅ a = 4 ⋅ 6 = 24 p=a+a+a+a=4\cdot a=4\cdot 6=24p=a +a +a +a =4 ⋅ a =4 ⋅ 6 = 2 4

We know the dozhina of apothem:

L = 3 ⋅ a = 3 ⋅ 6 = 18 l=3\cdot a=3\cdot 6=18l =3 ⋅ a =3 ⋅ 6 = 1 8

For our viewpoint:

S quad = S main S_(text(quad))=S_(text(main))S quad = S main

I lost my knowledge of the area of the bіchnі surfіnі. For the formula:

S bіk = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 18 ⋅ 24 = 216 S_(\text(side))=\frac(1)(2)\cdot l\cdot p=\frac(1)(2) \cdot 18\cdot 24=216S bik = 2 1 ⋅ l ⋅p=2 1 ⋅ 1 8 ⋅ 2 4 = 2 1 6 (Div. sq.)

Povna area:

$S_ (text (side)) + S_ (text (main)) = 216 +36 = 252$

Suggestion: 252 cm sq.

Instruction

Let us first, varto understand, that the bіchna surface of the pyramid is represented by dekіlkoma trikutniks, the areas of which can be known for the help of the most intriguing formulas, fallow in vіd vіdomi danih:

S \u003d (a * h) / 2 de h - height, lowered to bіk a;

S = a*b*sinβ, where a, b are the sides of the tricot, and β is the cut between the sides;

S \u003d (r * (a + b + c)) / 2, de a, b, c - The sides of the tricot, and r is the radius of the stake inscribed in the trikut;

S \u003d (a * b * c) / 4 * R, de R - the radius of the tricutnik described around the stake;

S \u003d (a * b) / 2 \u003d r² + 2 * r * R (like a tricutnik - rectangular);

S \u003d S \u003d (a² * √3) / 4 (like a tricot - equilateral).

In fact, there are no more basic formulas for knowing the area of the tricot.

Razrahuvav for help designate more formulas for the area of all tricots, which are the faces of the pyramid, you can proceed to the calculation of the area of \u200b\u200bthe pyramid. It’s even easier to fight: it’s necessary to lay down the area of \u200b\u200ball the tricots, which will make the surface of the pyramid. The formula can be said like this:

Sp \u003d ΣSi, de Sp - the area of \u200b\u200bbіchnoї, Si - the area of \u200b\u200bthe i-th tricot, which is part of the її bіchnї surface.

For greater clarity, you can look at a small butt: a regular pyramid is given, the boulder faces are made with equal-sided tricots, and the square is at the heart of it. The length of the edge of this pyramid should be 17 cm. It is necessary to know the square of the outer surface of this pyramid.

Solution: in the house the ribs of this pyramid are visible, in the presence of її between - equilateral tricots. In this way, it can be said that all sides of all tricots are 17 cm thick on the surface. Therefore, in order to loosen up the area of any of these trikuts, it is necessary to fill in the formula:

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

Apparently, there is a square on the base of the pyramid. In such a rank, apparently, that these are equal-sided trikutniks chotiri. Then the square of the bіchnoї surface of the pyramid is covered as follows:

125.137 cm² * 4 = 500.548 cm²

Expectation: the area of the stone surface of the pyramid is 500.548 cm²

On the back, I count the area of the bіchnі surface of the pyramid. Under the bіchnuyu surface, the area of \u200b\u200busіkh bіchnyh faces is rubbed on the surface. If you maєte to the right with a regular pyramid (that’s the one that is based on the correct bagatokutnik, and the top is projected into the center of that bagatokutnik), then to calculate all the bіchnі surface it’s enough to multiply the perimeter of the base lies (that’s the sum of the dozhins of the ) on the height of the bіchnі verge (іnakshe zvanої apothem) and dividing the otrimane value by 2: Sb = 1/2P * h, de Sb - the whole area of the bіchnі surfіnі, P - the perimeter of the base, h - the height of the bіchnі vаnі (apothem).

As if there is a pretty pyramid in front of you, it will happen to count the areas of all the faces, that fold them up. Scattered with beetroot faces of the pyramid are tricots, speed up with the formula of the area of the trikutnik: S = 1/2b * h, de b is the base of the trikutnik, and h is the height. If the area of all the faces is counted, it is not enough to fold them, so that the area of the outer surface of the pyramid should be taken away.

Then it is necessary to calculate the area of the base of the pyramid. The choice of the formula for the rozrahunka to lie down depends on which bagatokutnik lies at the support of the pyramid: correct (that is, on the other hand, it may be the same dovzhina) or incorrect. The area of a regular bagatokutnik can be calculated by multiplying the perimeter by the radius of the stake inscribed in the bagatokutnik and adding the value by 2: Sn \u003d 1 / 2P * r, de Sn is the area of \u200b\u200bthe bagatokutnik, P is the perimeter, and r is the radius of the stake inscribed in the bagatokutnik .

The truncated pyramid is a richly shaped one, which is set up by a pyramid and a lintel, parallel to the base. It is easy to determine the area of the bіchnі surface of the pyramid. It’s even simpler: the area is more expensive to get half of the sumi, substituting for apothema. Let's look at the butt of the rosewood square on the surface of the truncated pyramid. It is permissible, given the correct chotirikutna pyramid. Dovzhini dor_vnyuyut b = 5 cm, c = 3 cm. Apothem a = 4 cm. For a large pedestal, the veins are more p1=4b=4*5=20 cm. For a smaller pedestal, the formula will be offensive: p2=4c=4*3=12 cm. 4=32/2*4=64 div.

Pyramid - tse bagatohedron, one of the faces of this (base) is a dovilny bagatokutnik, the other faces (sides) are tricutniks, shaping the apex. For a lot of kutіv, the bases of the pyramid are trikutnі (tetrahedron), chotirikutnі thinly.

The pyramid is a bagatohedron, which has a base at the look of a bagatokutnik, and other faces are trikutniks from a sour top. Apothema is called the height of the bіchnі vіchnі vіrії prіramіdi, yak z її peaks.

- it is richly faceted to stand, the basis of which is a bagatokutnik, and the other faces are represented by trikutniks from the heady peak.

If the square is the basis, then the pyramid is called chotiricutny, like a trikutnik - then knitted. The height of the pyramid is drawn from її peaks perpendicular to the base. Also for rozrahunka area vikoristovuєtsya apothem- The height of the bіchnі verge, omitted from її peaks.
The formula of the square of the square of the surface of the pyramid is the sum of the square of її of the squares of the faces, yakі equal among themselves. However, this method of rozrahunka zastosovuetsya very rarely. In the main area of the pyramid, it is built up through the perimeter of the base and apothem:

Let's look at the butt of the rosewood square on the surface of the pyramid.

Let there be given a pyramid with a base ABCDE and a top F. AB =BC =CD =DE =EA =3 cm. Apothem a = 5 cm.
We know the perimeter. Oskіlki all the faces of the base are equal, then the perimeter of the p'yatikutnik is more expensive:
Now you can know the square of the pyramid:

The square of the regular tricot pyramid

The correct tricot pyramid is formed from the base, in which lie the correct tricot and three beetle faces, like equal squares.
The formula of the square of the square of the surface of the regular tricot pyramid can be protected by a different method. You can zastosuvat zvichaynu formula rozrahunka through the perimeter and apotheme, or you can know the area of one facet multiply її by three. If the edge of the pyramid is a tricot, then we will formulate the formula for the area of the tricot. For her, an apothem and a foundation will be needed. Let's look at the butt of the rosewood square on the surface of the regular knitted pyramid.

Given a pyramid with an apothem a = 4 cm and a facet of the base b = 2 cm. Find the area of the outer surface of the pyramid.
For the cob, we know the area of one of the bichny faces. I'll have a vipadku won't:
Substitute the value of the formula:
Since the right pyramid has all the sides of the same, then the area of the side surface of the pyramid is equal to the sum of the areas of three faces. Vidpovidno:

Square of the truncated pyramid

truncated a pyramid is called a bagatohedron, which is established by a pyramid and a peretina, parallel to the base.
The formula for the square of the square surface of the truncated pyramid is even simpler. The area is worth half the sum of the perimeters, substantiating the apothem:

Your privacy is important to us. For reasons we have expanded the Privacy Policy, as described, as we have collected your information. Be kind, read our privacy policy and let us know if you have any questions about food.

Selection of selected personal information

Under the personal information, data are given, as it is possible to win for the identification of a singing individual and a link with him.

You may be asked for your personal information if you contact us.

Below, there are some examples of types of personal information, as we can choose, and as we can select such information.

How we collect personal information:

If you submit an application on the site, we can collect various information, including your name, phone number, email address, etc.

How we collect your personal information:

The personal information collected by us allows us to contact you and tell you about unique offers, promotions and others, visit and find the closest ones.
From time to time we can vikoristovuvat your personal information to strengthen important reminders and reminders.
We can also collect personal information for internal purposes, such as auditing, analyzing data and other records with a method of improving services, which we hope is given to you by recommending our services.
As you take part in prize draws, competitions, or similar incentive entry, we can win information, hopefully, to manage such programs.

Disclosure of information to third persons

We do not disclose your information to third persons.

Vinyatki:

It is necessary - according to the law, judicial order, judicial review, and / or on the basis of public requests or requests from state authorities on the territory of the Russian Federation - disclose your personal information. We can also reveal information about you, even more importantly, that such disclosure is necessary or appropriately for safety, maintaining law and order, or other important vipadkiv.
In times of reorganization, aggravating or selling, we can transfer personal information that is collected by us, the third person - to the offender.

Protector of personal information

We live abroad - including administrative, technical and physical - for the protection of your personal information in the form of waste, theft and unscrupulous vikoristannya, as well as unauthorized access, disclosure, change of that infringement.

Maintaining your privacy at a peer company

In order to change your personal information so that your personal information is kept safe, we bring the norms of confidentiality and security to our contacts, and we strictly follow the rules of confidentiality.

How to know the surface of the pyramid formula. The square of the Chotirikutnoi Pyramid. The surface area of ​​the tricot pyramid