Yak to know the area of the basis of the chotirikutnoy pіramіdi. The area of the bichesky surface of the correct chotirikutnoy pyramidi: formulas and butts of problems
Instructions
The first for everything, varto vividness, that the bichny surface of the pirida is represented by the decilkom of tricytes, the areas of which you can know for the help of new formulas, in the abundance of all kinds of data:
S = (a * h) / 2 de h - hanging, lowered to bik a;
S = a * b * sinβ, de a, b - sides of the tricot, and - kut between the sides;
S = (r * (a + b + c)) / 2, de a, b, c are the sides of the tricycle, and r is the radius inscribed in the tricut of the stake;
S = (a * b * c) / 4 * R, de R - the radius of the tricutnik described at about the stake;
S = (a * b) / 2 = r² + 2 * r * R (yaksho tricut - rectangular);
S = S = (a² * √3) / 4 (like a tricycle - one-sided).
For the sake of it all, there are basic formulas for the known area of a tricycle.
Having rooted for the additional meaning of the formulas of the area of all tricycles, with the edges of the pyramid, it is possible to proceed to the calculation of the area of the pyramid. Fighting is even simpler: it is necessary to lay down the areas of all tricycles, so that you can set the surface of the piracy. The price formula can be displayed like this:
Sп = ΣSi, de Sп - area of the bichoi, Si - area of the i-th tricycle, but partly of the bіchnoi surface.
For greater clarity, you can look at a small butt: given to the correct pyramid, bichny faces which are set by unilateral tricycles, and a square is the basis of it. Dovzhin's ribs given to the pyramid become 17 cm. It is necessary to know the area of the natural surface of the pyramid.
Solution: from the house to the ribs of the given pyramid, from the house to the edge - the one-sided trikutniki. In such a rank, it is possible to say that all sides of all tricycle surfaces should be 17 cm long.
S = (17 ² * √ 3) / 4 = (289 * 1.732) / 4 = 125.137 cm ²
Seemingly, the square is at the base of the paramid. Such a rank, seemingly, is given by the mono-sided trikutniks of the chotiri. Todi of the area of the bicheskoy surface of the piracy should be secured as follows:
125.137 cm² * 4 = 500.548 cm²
Suggestion: the area of the bichesky surface of the pyramid becomes 500.548 cm ²
The collection is numbered on the area of the bicho surface of the pyramid. From the bichesky surface to rely on the sum of the areas of the bichnyh faces. As long as I can make a right with the correct pyramid (so it’s like this, in the base of which is the correct bagatokutnik, and the top is designed to the center of the bagatokutnik), then to enumerate all bichesky surfaces it is enough to increase the perimeter of the base ) on the height of the bichesky face (the name of the apothem) and the division of the value by 2: Sb = 1 / 2P * h, de Sb - the whole area of the bicon's surface, P - the perimeter of the base, h - the height of the bichesky side (apothem).
As soon as you are in front of you, there is a great piracy, it will be possible to count the areas of all the edges, that will be folded. Oskіlki bichny edges of the tricytny tricytnik, speed by the formula of the tricytnyk area: S = 1 / 2b * h, de b is the base of the tricytnik, and h is the height. If the area of all the edges is numbered, if you lose the folds, then trim the area of the bichesky surface of the pyramid.
Therefore, it is necessary to calculate the area of the base of the pіramіdi. Vibir of the formula for rozrakhunku lie in the fact that a bagatokutnik lie in the basis of a paramid: correct (that is, both sides of which may be the same for the same reason) and incorrect. The area of the correct bagatokutnik can be calculated by multiplying the perimeter by the radius of the stake inscribed in the bagatokutnik, and by adding the value to 2: Sn = 1 / 2P * r, de Sn - the chain of the bagatokutnik, P - the center of the perimeter, and r - the central radius of the inscribed ...
Accelerated piramida - tse bagatogrannik, which is set up as a piracy and peretin, parallel to pіdstavі. It is much easier to know the area of the bicheskoy surface of the piracy. Її is even simpler: the area is large enough to add half of the sumi before it. The butt is easy to see and the surface of the surface is large. It is possible that the correct pyramid is given. Dovzhini bases dorіvnyuyut b = 5 cm, c = 3 cm. Apothem a = 4 cm. To know the area of the common surface of the pyramid, it is necessary to know the perimeter of the bases. At the great pavilion, the road is p1 = 4b = 4 * 5 = 20 cm. For the young man, the formula will be offensive: p2 = 4c = 4 * 3 = 12 cm. 4 = 32/2 * 4 = 64 div.
Tricut piracy to be called a tricycle, which is based on the correct tricycle.
At such pіramіdі faces, the bases and edges of the bіchny sides are equal to ourselves. As a matter of fact, the area of the bichy faces is overlapped from the sum of the areas of three identical triplets. It is possible to know the area of the biological surface of the correct pyramid by the formula. And it is possible to grow a rosary at a kilka developed shvidshe. For the whole, it is necessary to establish the formula for the area of the biological surface of the triangular piracy:
de p - the perimeter of the base, at which all sides have b, a - apothem, dropped from the top to the center of the base. The butt of the tricot square is easy to see.
Zabdannya: Nekhai given the correct piracy. The side of the tricycle, which is to lie at the front door, b = 4 cm.
Oscillations behind the minds of the staff know all the necessary elements, we know the perimeter. Remember, for the correct tricycle, the sides should be equal, the perimeter will be secured according to the formula:
By the way, this is known to mean:
Now, I know the perimeter, we can rozrakhovuvat the area of the bichesky surface: 
To find out the formula for the area of the tricot pyramid for the calculation of the total value, it is necessary to know the area of the base of the polygon. For tsyogo vikorystyutsya formula: 
The formula of the area of the base of the tricot pyramid can be used. It is allowed to store any parameters for a given figure, but most often it is necessary. The butt is easy to see and the area of the base of the tricot pirida is visible.
Manager: The right side of the tricycle is at the right side, which should lie in the base, the side is a = 6 cm. Root the base area.
For the calculation, we need to deprive the party of the correct tricycle to get rid of it at the front of the party. By the way, given the formula: 
It is often necessary to reach the delivery to know about the area of the bagatogrannik. For the whole need to flatten the surface of the base surface. 
The butt of the tricot square is easy to see.
Zavdannya: do not give a correct trikutna pіramіda. The side of the base is b = 4 cm, apothem a = 6 cm.
For a cob, we know the area of a small surface for the same formula. Rosrahuєmo perimeter:
Pidstavlyaєmo dana formula: 
Now I know the base area: 
Knowing the area of the base of that common surface, I know the area of the ground: 
When designing the area correctly, do not forget about those that are based on the correct tricycle and the rich elements of the rim of the back.
- tse post, the basis of which is a great bagatokutnik, and many faces are presented by tricytails. These peaks lie at the same point and show the peaks of the paramid.
Piramida can be versatile - tricot, chotirikut, six-loop meager. I can name it, there is a significant amount of deposition from the number of kutiv, which will lie down to the base.
Correct pіramіda to be called pirama, in which the sides are the bases, the cut and the ribs. Similarly, in such piracy, there are areas of bichy edges.
The formula for the area of the bichesky surface of the piracy is the sum of the area of the edges:
In order to develop the area of the bicheskoy surface of the present pyramid, it is necessary to know the area of the skin okremogo tricycle that folds between you. Yakshu pіramіda has been strengthened, її between ії is represented by trapezіy. For the correct formula. At a certain area of the common surface, it is necessary to secure the base through the pivperimeter of the base and to the end of the apothemi:
The butt is easy to see, the area of the bichoi surface is clear.
Nehay is given the correct chotirikutna piramida. Side of falling asleep b= 6 cm, and apothem a= 8 cm
In the basis of the correct chotirikutnoy pyramid lie a square. For the cob, the perimeter is known:
Now we can prorahuvat the area of the bichesky surface of our piracy: 
In order to know about the area of the rim, it is necessary to know the area of the base. The formula of the area of the base can be seen, it is fallen from the fact that a bagatokutnik lie in the base. For many vicarious formulas of the area of the tricycle, parallelogram areas etc.
Look at the butt for the size of the base of the piracy, given by our minds. Oskilki pіramіda is correct, at the її base lie a square.
Square area insurance for the formula:,
de a - Side of the square. We have a size of 6 cm. 
Now I don't need to know about the square of the edge. The formula for the area of \ u200b \ u200bpirady is stored from the sum of the areas of the base and the base surface.
Viznachennya. Bichna edge- Tse a tricycle, for which one kut lies at the top of the pirudin, and the other side of the other lies on the side of the base (bagatokutnik).
Viznachennya. Bichni ribs- the whole side of the side edges. At the pyramid of the style of the ribs, the cut of the grooves at the bagatokutnik.
Viznachennya. Visota pіramidi- Tse perpendicular, drooping from the top to the base of the pіramіdi.
Viznachennya. Apothem- The perpendicular to the bichesky side of the pyramid, the descents from the top of the pyramid to the side of the base.
Viznachennya. Diagonal transition- tse pererіz pіramіdi area, so pass through the top of the pіramіdi and diagonal base.
Viznachennya. Correct pіramіda- tse pіramіda, in yaky base є the correct bagatokutnik, and the height goes down to the center of the base.
The volume of that area of the surface of the pyramid
Formulas. Ob'am pіramidi across the base area:
Power of power
If all the ribs are small, then near the base of the pyramid it is possible to describe the colo, and the center of the base is located behind the center of the stake. Also, the perpendicular, drooping from the top, pass through the center of the base (stake).
Yakshcho bichnі ribs of івні, all the stench healed up to the area under the same kutas.
Bichni ribs of the ryvnі todі, if the stench is set from the area of the base of the rіvnі kuti, as well as near the base of the pyramid can be described colo.
As long as there are many faces nahileni to the area of the base under one cut, then the base can be entered into the base, and the top is designed to go to the center.
As many facets are nailed to the surface as one cut, then apothem of ordinary faces.
Power is correct
1. The top of the pіramіdi rіvnovіddalen is along the main paths.
2. The strongest ribs of the ribs.
3. Stretch the ribs with the same cut to the base.
4. Apofemi of all biches of the rivn.
5. The area of all bichny edges of the ryvn.
6. Both sides may be of the same two-sided (flat) kuti.
7. Around pіramіdi it is possible to describe the sphere. The center of the described sphere will be the point of perpendicular overflow, which passes through the middle of the ribs.
8. Up to the time you can write a sphere. The center of the inscribed sphere will be the point of intersection of the bisectors, which goes from the cut between the edge and the base.
9. If the center of the inscribed sphere is located at the center of the described sphere, then the sum of flat kutiv at the tops of the road π or navpaki one kut is expensive π / n;
The link of the pіramіdi with the sphere
Around the piracy, it is possible to describe the sphere of that, if at the base of the pyramid there is a large hedgeon near which it is possible to describe a colo (a sufficient amount of mind is required). The center of the sphere will be the point of the cross-flow of the areas, which will pass perpendicularly through the middle of the side edges of the pramid.
Around whether it is tricky and correct, it is possible to describe the sphere.
At the stage it is possible to enter a sphere, as the bisectorial areas of the inner two-sided cuts can be merged into one point (a sufficient amount of intelligence is required). Qi point will be in the middle of the sphere.
Pіramіdi link with a cone
The cone is called inscribed at the piracy, when the vertices are scattered, and the base of the cone is inscribed in the base of the piracy.
The cone can be inscribed up to the pyramid, as apofemi the pyramid rivni between oneself.
The cone is called described near the pyramid, where the vertices are swayed, and the base of the cone is described near the base of the pyramid.
The cone can be described near the middle, as well as all the ribs along the middle.
Linkage of pyramid with a cylinder
The pyramid is called inscribed in the cylinder, as the top of the pyramid lies on the same base of the cylinder, and the base of the pyramid is inscribed in the base of the cylinder.
The cylinder can be described near the paramid, as well as near the base of the paramid can be described colo.
At the chotiragon, the chotiri faces and the chotiri tops have the same number of ribs, if only two ribs do not conceal the spiral peaks and do not stick.
The skin top is folded from three edges and edges, which are trigonal cut.
From the center of the other side to be called the median of the chotiragon(GM).
Bіmedіanoyu to be called as if from the middle of the protruding ribs, which do not stick (KL).
The biomedia and the median of the chotiragon are interlaced in one point (S). When there is a lot of media, it’s going to move, and media is going to be 3: 1, we’re fixing from the top.
Viznachennya. GOSTROKUTNA PIRAMIDA- tse pіramіda in yakіy apothem is more than half of all sides of the base.
Viznachennya. Tupokutna pіramida- tse pіramіda in yakіy apothem is less than half of all sides of the base.
Viznachennya. Correct tetrahedron- chotiragon, at which all chotiri faces are equilateral trikutniks. Win - one of five correct bagatokutniks. In a regular tetrahedron, all dihedral kuti (with edges) and trigonal kuti (at the vertex) are equal.
Viznachennya. Rectangular tetrahedron to be called a chotiragon at a straight line between three ribs at the tops (ribs are perpendicular). Three facets to make rectangular trikutny kutі faces є rectangular tricycles, and the base є full tricycle. Apothem be-like between the two sides of the base, like the fall of apothem.
Viznachennya. Equal tetrahedron to be called a chotiragon at a certain number of faces of the same person, and the basis is a regular tricycle. Such a tetrahedron has tricycle edges.
Viznachennya. Orthocentric tetrahedron to be called a chotiragon, at which all the height (perpendiculars), which is lowered from the top to the opposite edge, melt into one point.
Viznachennya. Zirkova piramida to be called rhinestone, in which the base is є zirka.
Chi є is the formula? Nі, zagalі dumb. It is just necessary to shukati the areas of the bichy edges and pidsumovuvati їkh.
The formula can be written for straight prisms:
De - the perimeter of the base.
But nevertheless, it’s simpler for the skin data to drop the skin of all areas, not to remember the additional formulas. On the surface, I will turn the surface with correct six-cut prisms.
The most common faces are rectangular. To mean.
Tse vzhe vivodili for a pidrakhunka obsyagu.
Otzhe, otrimo:
The area of the surface
For piradi, there is also a general rule:
Now let’s cover the surface of the most popular pyramids.
The area of the surface of the correct tricot pyramid
Leave the side to the base of the road, and the edge to the bottom. Need to know in.
Guess now, scho
Tse the area of the correct tricycle.
I still guess, yak shukati tsyu area. Vikoristovuєmo area formula:
We have "" - tse, and "" - chain, and.
Now we know.
Corrode the basic formula of the area and the Pythagorean theorem, we know
Uvaga: if you have the correct tetrahedron (tobto), then the formula is as follows:
The area of the surface of the correct chotiricut pyramid
Leave the side to the base of the road, and the edge to the bottom.
Basically - a square, and that.
It was too late to know the area of the biches
The area of the surface is correct six-sided pyramid.
Let the side be the base of the door, but the bottom edge.
Know yak? A six-cycle tracker is stored from six identical correct jersey pieces. The area of the correct tricycle was already whispered when the area of the surface of the correct tricycle was pitted, here the victorious formula is known.
Well, and the area of the bichy faces was already joked as many as two
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