Yak know bichnu. How to calculate the area of ​​\ u200b \ u200bpirady: present, bichnu and again? Pіramidi tie with a cone

Enter a few sides, one side and one apothem:

viznennya pіramіdi

piramida- tricycle, at the base of which is the tricycle, and between the two - tricycles.

Online calculator

Varto zupinitiya on viznachennі deyakie warehouse pіramіdi.

At her, yak and at the ones of the dwarfs, є ribs... Stinks converge to a single point, yak to be called top pіramіdi. In the її basis can be a good bagatokutnik. border be called geometric figure, Approved by one side of the presentation and two by the closest ribs. Our vipadku has a tricutnik. hanging pіramіdi be called vіdstanі from the area, in which lie її before, up to the top of the bagatogranik. For the correct understanding apofemi- the perpendicular, the descents from the top of the piridi to the її base.

vidi pіramid

Find 3 types of pirits:

1. rectangular- the one, if you have an edge, I will fix it straight kut with the base.
2. correct- she has a correct geometric figure, and the top of the bagatokutnik itself is projected towards the center of the street.
3. tetrahedron- pіramіda, folded from trikutnikіv. Moreover, the skin of them can be accepted as a basis.

Formula of the surface area of ​​the piracy

For a common area of ​​the surface of the surface, it is necessary to use the area of ​​the surface of the individual surface and the area of ​​\ u200b \ u200bpidstavi.

By the simplest and by the correct piracy, we are engaged in it. Numerously I will increase the area of ​​the surface of such a pyramid. The area of ​​\ u200b \ u200bbichnoy surface of the road:

S-pl_ch = 1 2 ⋅ l ⋅ p S _ (\ text (side)) = \ frac (1) (2) \ cdot l \ cdot pS pl_ch​ = 2 1 ​ ⋅ l ⋅p

L l l- apothem pirady;
p p p- the perimeter of the base of the pyramid.

The total area of ​​the surface of the pyramid:

S = S bik + S main S = S _ (\ text (side)) + S _ (\ text (main))S =S pl_ch​ + S main​

S-plich S _ (\ text (side)) S pl_ch​ - the area of ​​the bichnoy surface of the pyramid;
S main S _ (\ text (main)) S main​ - the area of ​​the base of the pіramіdi.

Application of solving tasks.

butt

To know about the area of ​​the tricot piracy, which is the apothem of the road 8 (div.), And in the base lie the equal tricycle on the side 3 (div.)

Decision

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

We know the perimeter of the building. So, as a basis, lie a single-sided trikutnik with a side a a a, Thats yogo perimeter p p 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 piradi area:

S-plіch = 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 pl_ch​ = 2 1 ​ ⋅ l ⋅p =2 1 ​ ⋅ 8 ⋅ 9 = 3 6 (Div. Sq.)

Now we know the area of ​​the base of the pіramidi, that is, the area of ​​the tricycle. In our vipad, the tricycle is equal to the і th area can be calculated by 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- side of the tricot.

otrimuєmo:

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 bik + S main ≈ 36 + 3.9 = 39.9 S = S _ (\ text (side)) + S _ (\ text (main)) \ approx36 + 3.9 = 39.9S =S pl_ch​ + S main​ ≈ 3 6 + 3 . 9 = 3 9 . 9 (Div. Sq.)

as follows: 39.9 cm. Sq.

Another butt, three folds.

butt

Pіdstavu pіramіdi є square with area 36 (see sq.). Apothem of Bagatogrannik 3 times larger than the side of the base a a a... I will know about the area of ​​the surface of the given figuri.

Decision

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

We know the side of the side, tobto the side of the square. Yogo area and on the side and tied:

S quad = a 2 S _ (\ text (quad)) = a ^ 2S quad​ = a 2
36 = a 2 36 = a ^ 2 3 6 = a 2
a = 6 a = 6 a =6

We know the perimeter of the base of the pіramidi (tobto, 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 about the genie of apofemi:

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

Our vipad:

S quad = S main S _ (\ text (quad)) = S _ (\ text (main))S quad​ = S main​

It’s too late to know only the area of ​​the bichesky surface. For the formula:

S-plіch = 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 pl_ch​ = 2 1 ​ ⋅ l ⋅p =2 1 ​ ⋅ 1 8 ⋅ 2 4 = 2 1 6 (Div. Sq.)

Povna area:

$S=S^{\text{pl\_ch + Smain = 2 1 6 + 3 6 = 2 5 2 (Div. Sq.)}}$

as follows: 252 cm. Sq.

Typical geometric landmarks on the area and in the trivial space є the problem of the size of the area on the surface of the small figures. At the given statti, the formula of the area of ​​the biological surface is correct.

What is pirama?

Guided by Suvore geometrically pіramіdi. Supposedly, є a good bagatokutnik with n sides and with n kutas. Viberemo will allow the point to be spacious, as it will not be known in the area of ​​the designated n-kutnik, but at the same time with the skin top of the kutnik. Mi otrimaєmo figuru, scho a singing obsyag, yak is called n-vugіlо pіramіda. For the butt, it is shown on the bottom of the little one, like a viglyadє five-square piremida.

Two important elements of being a piracy are the base (n-kutnik) and the top. There are one element for one with one n tricycle, as in a zagalny vipad not carry one to one. The perpendicular, drooping from the top to the base, is called the visota of the figuri. As soon as I move to the geometric center (to be located at the center of the bagatokutnik), then I call it straight. Yaksho krym tsyogo umovi pіdstav є correct bagatokootnik That is, the whole time is called correct. The little ones will show below how they look at the correct pirits with tricky, chotirikutny, five-and six-minute pisces.

the surface of the pіramіdi

First, go to the food about the area of ​​the bichesky surface of the correct pyramid of the chotiricut, then the reports are made on the understanding of the surface itself.

Yak bulo said vishche i is shown on the little ones, whether the pirimida is set by a set of faces abo sides. One side є before, і n sides є trikutniki. The top of the whole figuri is the whole sum of the area of ​​the skin side.

The upper part is manually screwed onto the butt of the figuri rosette. The rosette for the correct chotirikutnoy pyramid is shown on the baby below.

Bachimo, which is the area of ​​the surface of the road, the area of ​​the same area of ​​the same tricycle and the area of ​​the square.

The home area of ​​all tricycles, which fixes the biches of the figurines, is taken to be the biches of biches. It’s possible to show it, as it’s possible to make it right for the right thing.

The area of ​​the bichesky surface of the chotirikutnoy correct pyramidi

Schob count the area of ​​the bichesky surface of the designated figuri, which is known to be fertile before the induced rozgorttsi. Supposedly, we see the side of a square display. Significant її symbol a. It can be seen that it is leather from chotiroh of the same tricycles, but now a. To count the total area, it is necessary for the nobility to measure the value for one tricycle. In the course of geometries of the city, where the tricycle of the area S t dorіvnyu dobutku base on the height, like the slіd podіliti navpіl. Tobto:

De h b - visota tricycle tricycle, Held up to the basics a. For pіramіdi tsya visota є apotema. Now it is necessary to multiply the otrimation of the viraz by 4, but to trim the area S b of the surface for the given pyramid:

S b = 4 * S t = 2 * h b * a.

The formula is to take revenge on two parameters: APVT and the submission side. As soon as I remain in the greatest minds of the head of the house, then I will be able to count, I know the magnitude. Above all, formulas for rosrahunku apotemi h b for two types:

• if there is more than a bicon rib;
• if the house is visible.

Yakshto denote the edge of the lateral rib (side of the tricot tricot) by the symbol L, so apotema h b is based on the formula:

h b = √ (L 2 - a 2/4).

The price rotation is the result of storing Pyfagor's theorems for the tricycle of the bichniy surface.

Whenever there is a height h piradi, todi APVT h b can be developed as follows:

h b = √ (h 2 + a 2/4).

The trimming of the hangs is also not foldable, as it is possible to look through the middle of the frame rectangular tricycle, Approvals with legs h і a / 2 і hypotenuse h b.

It will be shown that you have fixed the formulas that you have done tsikavi zavdannya.

Zavdannya z vіdomuyu area of ​​the surface

Seemingly, the area of ​​the bichesky surface of the chotirikutnoy road is 108 cm 2. It is necessary to calculate the value of the second apotemi h b, as the height of the road is 7 cm.

We can write down the formula for the surface area S b through the height. maєmo:

S b = 2 * √ (h 2 + a 2/4) * a.

Here we just submitted an approximate apotemi formula to viraz for S b. We make offenses of parts of equality in a square:

S b 2 = 4 * a 2 * h 2 + a 4.

To know the meaning of a, I will completely replace the winners:

t 2 + 4 * h 2 * t - S b 2 = 0.

Nowadays, we see the meanings and the most square ones:

t 2 + 196 * t - 11664 = 0.

We wrote only the positive root of the ryvnyannya. Todi Sides Basics Piridi Bude Dorivnyu:

a = √t = √47.8355 ≈ 6.916 cm.

Schob otrimati dozhinu apotemi, finish quickly with the formula:

h b = √ (h 2 + a 2/4) = √ (7 2 + 6.916 2/4) ≈ 7.808 cm.

Bichna surface of the piracy Cheops

Visually significant for the greatest Egyptian piracy. Seemingly, in the її base lie a square on the back side of 230.363 meters. The visota and spatka became 146.5 meters. With the help of the numbers in the similar formula for S b, we can recognize:

S b = 2 * √ (h 2 + a 2/4) * a = 2 * √ (146.5 2 +230.363 2/4) * 230.363 ≈ 85 860 m 2.

Know the value of three more than 17 football fields.

piramida- one of the types of bagatogranits, set with bagatokutniks and trikutniks, which lie in the base and with its edges.

Moreover, at the tops of the piracy (tobto in one point), all the faces are united.

In order to calculate the area of ​​the piracy, it is possible to calculate the size, and the lateral surface should be stored from decilkoh tricycles. And it’s easy to know that

різні formulas. This is due to the fact that we see the tricky dynasty, mi shukamo їkh area.

Pererakhumo deyakі formulas, for the help of which you can know the area of ​​trikutnikіv:

1. S = (a * h) / 2 ... In the given vipadku we see the height of the tricycle h , Yaka is lowered to the side a .
2. S = a * b * sinβ ... Here are the sides of the trikutnik a , b , And kut mіzh them - β .
3. S = (r * (a + b + c)) / 2 ... Here are the sides of the trikutnik a, b, c ... Radius cola, yaka is inscribed in the trikutnik - r .
4. S = (a * b * c) / 4 * R ... Radius of the described stake near the trikutnik - R .
5. S = (a * b) / 2 = r² + 2 * r * R ... I have given the formula it is necessary to fix it only in that case, if the trikutnik is rectangular.
6. S = (a² * √3) / 4 ... Qiu formula zastosovuєmo to an equal-sided trikutnik.

If only for that, as the area of ​​all trikutniki, as the facets of our pyramid, can be counted as the area of ​​the bichniy surface. For all, we will be victorious over the formulas.

In order to calculate the area of ​​the common surface of the pyramid, there are no folding areas: it is necessary to know the sum of the areas of all the triplets. Vislovimo tse formula:

Sп = ΣSi

here Si є the area of ​​the first tricot, and S NS - the area of ​​the bichnoy surface of the pyramid.

Visible on the butt. Given the correct piramida, її bіchnі boundaries approved by decіlkom unilateral trikutniki,

« Geometry є hiring the best for the cleverness of our rosy health».

Galileo Galilei.

and the square є before the time. Moreover, the edge is 17 cm long.

Mіrkuєmo like this: we see it with the faces of the pіramіdi є trikutniki, the stench is equilateral. We also see, like a dozen ribs at the center of piracy. Come on, all tricycles may be on the same side, and even 17 cm.

For the calculation of the skin area of ​​tricycles, you can use the following formula:

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

So, as it is known, that the square lies in the basis of the piracy, then go, as it is, with the help of the equilateral trikutnik. And tse means that the area of ​​the bichesky surface of the piracy can be easily recovered for such a formula: 125.137 cm² * 4 = 500.548 cm²

Our type of offensive: 500.548 cm ² - this is the area of ​​the common surface of the pyramid.

At the correct trikutny pіramіdі SABC R- middle of the rib AB, S- top.
Vidomo scho SR = 6, And the area is bichesky surface of the road 36 .
Know about the dinner BC.

Zrobimo armchair. At the correct pirimidi bichni facet is a trikutnik.

vidrizok SR- median, lowered to the base, and hence, і the height of the bіchnoі faceі.

The area of ​​the bicheskoy surface of the correct tricyte pirimidi dorivnyu sumy area
three priests bichnyh faces S-pl_ch. = 3 S ABS... Zvidsi S ABS = 36: 3 = 12- face area.

The area of ​​the tricycle is in the back half of the house.
S ABS = 0.5 AB SR... I know the area and the height, we know the bik pіdstavi AB = BC.
12 = 0.5 AB 6
12 = 3 AB
AB = 4

vidpovid: 4

You can go to the factory and from the beginning of the day. Come on, show the side AB = BC = a.
Todi face area S ABS = 0.5 AB SR = 0.5 a 6 = 3a.

The area of ​​skin with three sides of the door 3a, The area of ​​three sides of the door 9a.
For the cleaning of the area of ​​the biches of the surface of the road 36.
S-pl_ch. = 9a = 36.
Zvidsi a = 4.

The area of ​​the bichesky surface of the modern piracy of the urban sum of the biches of the biches of the sides. A special formula for turning the whole area of ​​\ u200b \ u200bmay sense date at the same time. So, don’t give a correct piracy, the basis of which is the correct n-kutnik on the side, which is important. Nekhai h - the height of the bichesky side, also called apothem pіramіdi. The area of ​​one common facet is 1 / 2ah, and the entire lateral surface of the piracy is less than the area, equal to n / 2ha. So na is the perimeter of the base of the piracy, then you can write the known formula in the viewer:

The area of ​​the bichnoy surface Correctly send the transport to the apothemi half of the perimeter of the door.

Scho feel surface area, It’s just to the point of being presented.

Inscription and description of the sphere and cool... It is necessary to note that the center, inscribed in the paramid of the sphere, lies on the cross-section of the bisector area of ​​the inner two-edged cuts of the paramid. The center of the described sphere near the perpendicular to the spherical is to lie on the cross-section of the squares, which passes through the middle of the ribs of the perpendicular and perpendicular.

The piracy has been increased. As soon as the development of the area is parallel to the base, then the part, which is located between the common area and by the way, is called increased piracy. On the little one is shown a piracy, a vidkidayuchi part, how to lie in a small area, we can recognize a piracy. It is clear that it is small that the pyramid is homotetic with the center of homothety at the vertex. Coefficients for road transport heights: k = h 2 / h 1, for either side ribs, or for some kind of liner size for both sides. We know that the areas of small figures are to be carried, like the squares of the lines; so the area before both the piracy

Here S 1 is the area of ​​the lower base, and S 2 is the area of ​​the upper part of the truncated pyramid. At such a common there are many surfaces of the parasites. See also the rule for the parties.

Obsiagi podibnykh til to be rendered, like cubes of the lineworks; for example, when you send us messages, as you create them on the squares beforehand, it’s our rule to go right away. It’s an absolutely gallant character and is straightforward because of the fact that you will get the majestic dimension of the third stage of the genie. Use the cinnamon rule, vivedemo the formula, how can you obsyag increased piracy through the height and area of ​​the road.

Do not give an increase in size with height h and areas for S 1 and S 2. As soon as it has not been extended to the rest of the time, then the efficiency is full of bread and it is easy to know a little. The frequency of the accelerated pyramid twists and turns yak h = h 1 - h 2 = h 1 (1 - k). Now it’s a matter for the obsyagu of the increased piracy (through V 1 and V 2 of the signified obsyagu more and more little piracy)

formula obsyagu

We show the formula of the area S of the bichesky surface of the correct visualized pyramid through the perimeter of P 1 and P 2 until the end of the apofemia. Mіrkuєmo is exactly the same, as in the case of vivedenny formulas for ob'єmu. In addition to the upper part, maєmo P 2 = kP 1, S 2 = k 2 S 1, de k is the function of the requirements, P 1 and P 2 are the perimeter of the unit, and S 1 and S 2 are the children on the surface of all the upper part is exemplary. For any surface, we know (a 1 and a 2 - apofemi pyramid, a = a 1 - a 2 = a 1 (1-k))

the formula of the area of ​​the biological surface is correct