The area of ​​the base of the tricot tricycle is formula. The yak knows and the area of ​​the tricot (formulas)


Rise not to deprive the schoolchildren of the students, but of the real, practical life. For example, it’s necessary to improve the façade part before the hour. How much material is needed?

Often, there are majstri, which work from the fabric of the fabric, with the help of the staff. Even a lot of details, which are required for a master, can be used to create the shape of a tricot tricot.

Otzhe, there are many ways to help you to know the area of ​​the tricot tricycle. The first one is enumerated at the exhibition and at the top.

For the first time we need to find the MNP tricycle with the MN base and the PO height. Now you can do it in a chair: from the point P draw a line parallel to the base, and from the point M - a line parallel to the height. The point is called Q. I’m sure that I know the area of ​​the tricycle tricycle, it’s necessary to look at the MOPQ tricycle tricycle, on the other hand, on the other side of the MP tricycle tricycle.

I’ll get it right, I’m a straight-cutter. Oskіlki mi will be yogo samі, of course, both sides MO and OQ are parallel. The 1st side of QM and OP may be parallel. The POM cut is straight, and the OPQ can be straight. Otzhe, chotirikutnik, scho viyshov, є rectangular. It is difficult to find out which area is not in the warehouse; OM is half the base of this MPN tricycle. Looks like a viplive, which is the area of ​​the rectangular tricycle, which we have prompted, is built on top of the base.

Another stage of the set before us is a tricycle, which is due to the area of ​​the tricycle, є is proof of the fact that we deny the rectangular tricycle behind the area, so that, as the area of ​​the tricot, it is also worthy to the base.

Possibly for a cob tricot PON and PMQ. The stench of an offense is straightforward, so there is a straight kut in one of them of the statements of the head, and straight kut in the inside of a rectangle. Hypotenuzi at those with є sides of the femoral tricycle, also the same. The legs of the PO and QM are also parallel to the sides of the rectangle. Otzhe, the area of ​​the PON tricycle, and the PMQ tricycle is equal to oneself.

The area of ​​the rectangular QPOM goes to the areas of the trikutniks PQM and MOP in the sum. Having replaced the overdue of the QPM tricycle with the PON tricycle, we will accept it in the summaries given to us for the introduction of the theorem by the Tricutnik. Now we know how to know the area of ​​a tricot tricycle on the basis of and on the basis of - to count the size of it.

Ale it is possible to know, how to know the area of ​​the tricot tricycle from the base and sideways. There are also two options here: the theorem of Heron and Pythagoras. The solution is clear from the definitions of Pythagoras' theorems. For example, depending on the same PMN at the PO.

The rectangular tricycle POM MP has hypotenuse. Її square of doors sum of squares PO and OM. Oskіlki OM - half of the base, as we see, we can easily know OM and put the number square. Having made the output from the square of hypotenuse, the number is not correct, for the reason why the square of the leg is larger, which is the height of the tricycle. Knowing about the difference and knowing the height of the rectangular tricycle, you can give a view on the set in front of us.

You just need to multiply the value by the basis and the result of the distribution of the navpil. Why would it be so easy to fix, we explained at the first option to prove it.

Buvak, it is necessary to revise the calculation from the side of that kutu. That is, it is known that the basis, the vicorist formula with sines and cosines, is known, and, again, we multiply the result and multiply the result.

It is always in the mind of a tricycle that one sees a few options in the known area. For example, for calculating the area of ​​a rectangular tricycle, the formula is S = a * b / 2 de a і b - the whole leg. If you need to know the area of ​​a tricycle tricycle, it is necessary to have a distance for two tiers of yoghurt and height. Tobto S = b * h / 2 de b is the base of the tricut, and h is the th hem.

Distant can be found in the area of ​​the femoral rectangular tricycle. Here, help is coming from the offensive formula: S = a * a / 2, de catheti "a" і "a" - obov'yat kayut but with the same values.

Likewise, we often find ourselves counting the area of ​​a tricycle tricycle. Vaughn perebuvaє for the formula: S = a * h / 2, de a is the side of the tricycle, і h is the thigh. For the tsієyu formula: S = √3 / 4 * a ^ 2, de a - side.

The yak knows and the area of ​​the rectangular tricycle

Do you need to know the area of ​​a rectangular tricycle, even if the tasks do not specify the size of two legs at once? Todi with the formula (S = a * b / 2) we can speed up directly.

A small number of available options for connection are clear:

  • If you don’t have a home for one leg, if you were given a hypotenuse of the other leg, then you’ll go up to the great Pythagoras and for his theorem (a ^ 2 + b ^ 2 = c ^ 2), you’ll be able to get the gin of an unattended leg, because of the victorious for a rosy one.
  • Also given one leg of one leg and the degree of nahil kut of the prototype youmu: it is known that the other leg is given the formula - a = b * ctg (C).
  • Given: one leg and the degree of one leg close to one: for the same leg of the other leg, the formula is a = b * tg (C).
  • I stop, given: kut і dozhina hypotenusi: numbered as many as in both legs, according to such formulas - b = c * sin (C) і a = c * cos (C).

The yak knows and is the area of ​​the femoral tricycle

The area of ​​a femoral tricycle can be much easier and quicker to know for the formula S = b * h / 2, but for the duration of one of the indicators, it is much faster. It is also necessary to visit the documents.

Possible options for the establishment:

  • Given: one meal of one side from the other side of the other and one of a basis. It is known through the Pyfagorian theorem to the visota, to the genus of the other catheti. For the wash, the base is divided into two, є leg, and from the home side - by the hypotenuse.
  • Given: the base is that kut between the side of that base. It is calculated by the formula h = c * ctg (B) / 2 height (not forgetting the bik "c" for two).
  • It is given: the visota is that kut, as if the statement was made on the basis and on the other side: there is a formula c = h * tg (B) * 2 for the value of the sign, that the result is multiplied by two. The distance is numbered.
  • Vіdoma: dovzhina lateral side and that kut, which pretended to be between him and that visota. Decision: victorious formulas - c = a * sin (C) * 2 і h = a * cos (C) for the meaning of the basis of that visoti, for which I am important to the area.

Yak know the area of ​​the rіvno-femoral rectangular tricycle

If all these are given, then according to the standard formula S = a * a / 2 I can calculate the area of ​​a straight-hip rectangular tricycle, if there are no indications in the tasks, then it is necessary to admit the additional ones.

For example: we do not see a dovzhini of both sides (mi pam'yataєmo, in a ryvno-femoral rectangular tricot of a stink of rіvnі), ale is given a dovzhina hypotenuzi. Often the Pythagorean theorem means the same sides "a" and "a". Pythagoras' formula: a 2 + b 2 = c 2. In the fall with a straight-backed tricycle, it will be transformed into taku: 2a ^ 2 = c ^ 2. To go, to know the leg "a", it is necessary to get the hypotenuse for the root z 2. The result of the decision and will be done on both legs of the straight-femoral rectangular tricycle. The distance is known to the area.

Yak know the area of ​​the tricycle tricycle

With the help of the formula S = √3 / 4 * a ^ 2, one can easily wirahuvati the area of ​​the equilateral tricycle. As long as the radius of the described stake is tricot, then the area can be known by the formula: S = 3√3 / 4 * R ^ 2, de R is the radius of the stake.

Instructions

Video on the topic

Beast to respect








Dzherela:

For the ear of home about the meaning. The leg is called the side of the rectangular tricycle, the yak is lying down to the straight kut (tobto the warehouse from the іnshy side of the kut 90 degrees). Dovzhini cathetіv understandable to start a and b. The size of the gostrikh kut_v of a rectangular tricycle, which can be extended to the legs, is called A and B by definition. Hypotenuse is the name of the bik of a rectangular tricycle, which is opposite to a straight kuta (so that it is located right next to a straight kuta, with the other sides of a tricut I will make a gostri kuti). Dovzhinu hypotenuzi is meaningful through s. Shukanu square is meaningful through S.

Instructions

Stay in the formula S = (a ^ 2) / (2 * tg (A)) at that vypadku, if you have tasks deprived of one of the legs (a), or also vіdomy kut (A), to support the th leg. The sign "^ 2" denotes a square.

Victory the formula S = (a ^ 2) * tg (B) / 2 d y vypadku, if you do not have tasks with one of the legs (a), but also the vidomy cut (B), which is attached to the leg.

Video on the topic

Dzherela:

  • "Book of Mathematics for Entrants to Higher Educational Institutions", ed. G.M. Yakovlev, 1982.

Rivnostegnovim vvazhaєtsya such a trikutnik, at which two sides of the ryvnі. The area of ​​a tricycle can be developed using several methods.

Instructions

Video on the topic

Beast to respect

Find the signs of a femoral tricycle:
1) At a femoral tricycle є 2 rivnі kuti;
2) The visota of the tricycle is getting rid of the median;
3) The visota of the tricytta bears on the yogo bisectrix;
4) The bisectrix of the tricycle is getting used to this median;
5) The femoral tricycle has 2 median rivnias;
6) The femoral tricycle has 2 lengths;
7) The femoral tricycle has 2 bisectors.

Dzherela:

  • tricycle area

One of the figures, which can be seen in the lessons of mathematics and geometry, is a trikutnik. Trikutnik - bagatokutnik, yakogo has є 3 peaks (kuta) and 3 sides; a part of the area, surrounded by three dots, in pairs with three dots. Isnu bezlіch zavdan, tied from the knowledge of the growth of the figures of the figuri. One of them - area... Reserved from the previous tributes, the number of formulas for the area trikutnik.

Instructions

As soon as you see it, you will see a party that was held on its side. trikutnik, Vicorize the formula S =? h * a.

As soon as you see one side of the tricycle of the same height, it is lowered to the side, multiply one side by one side, and distribute the result on two sides.

As long as you are in front of you a straight-cut tricycle, follow the lines of up to the length of your legs, so that the sides, which lie down to the straight cut. Multiply by the number of legs, and the result is reduced by two.

As soon as you can tell about the size of the cut between two tricycles, and you see the tricycles, then I know the area of ​​the tricot by the formula:

St = ½ * A * B * sinα, de St - the area of ​​the tricycle; A and B - dozhini side of the trikutnik; α - kuta, roasted between the sides.

S = 1/2 (AB + BC + AC) = p r.

Calculate the perimeter:

p = (5 + 7 + 10) = 11.

Insure the required value:

S = √ (11 (11-5) (11-7) (11-10)) ≈ 16.2.

Three points that unambiguously represent the tricutnik in the Cartesian coordinate system - the whole vertex. Knowing the position of the skin from the coordinate axes, it is possible to calculate the parameters of the flat figure, including the perimeter. area... The price can be created in a number of ways.

Instructions

Vikoristovyte Heron's formula for area size trikutnik... Have nіy zadіyanі razmіri three sides fіguri, to that count, fix it. Dovzhina of the skin side is guilty of making the root iz the sum of the squares of the dovzhin projections on the coordinate axes. We can denote the coordinates A (X₁, Y₁, Z₁), B (X₂, Y₂, Z₂) and C (X₃, Y₃, Z₃), and in the genie їх sides can be violated as follows: AB = √ ((X₁-X₂) ² + (Y₁ -Y₂) ² + (Z₁-Z₂) ²), BC = √ ((X₂-X₃) ² + (Y₂-Y₃) ² + (Z₂-Z₃) ²), AC = √ ((X₁-X₃) ² + (Y₁-Y₃) ² + (Z₁-Z₃) ²).

To simplify the rosters, enter a supplementary change - for example, a perimeter (P). Z, so half the sumi dovzhin of the sides: P = ½ * (AB + BC + AC) = ½ * (√ ((X₁-X₂) ² + (Y₁-Y₂) ² + (Z₁-Z₂) ²) + √ ((X₂-X₃) ² + (Y₂-Y₃) ² + (Z₂-Z₃) ²) + √ ((X₁-X₃) ² + (Y₁-Y₃) ² + (Z₁-Z₃) ²).

Crack area(S) according to Heron's formula - find the root from the perimeter on the basis of the difference between them and the skin from the sides. The zagal її can be written as follows: S = √ (P * (P-AB) * (P-BC) * (P-AC)) = √ (P * (P-√ ((X₁-X₂) ²) + (Y₁ -Y₂) ² + (Z₁-Z₂) ²)) * (P-√ ((X₂-X₃) ² + (Y₂-Y₃) ² + (Z₂-Z₃) ²)) * (P-√ ((X₁) -X₃) ² + (Y₁-Y₃) ² + (Z₁-Z₃) ²)).

For practical applications, you can manually use special calculators. Tse scripts, distributed on the servers of some sites, such as to create all the necessary space on the basis of coordinates, as entered in the form. One such service is that it’s not enough to explain that rim for skin croc. To that, if you want to be deprived of the final result, and not assessed by the zealous viglyad, go, for example, to the side of http://planetcalc.ru/218/.

In the form field, enter the skin coordinate of the skin from the vertices trikutnik- the stink here yak Ax, Ay, Az, etc. As a tricycle of tasks with two-sided coordinates, at the field - Az, Bz and Cz - write zero. At the "Accuracy of calculation" field, set the required number of signs written by the Komi, clicking with a bear

    Z'yasuyte, how to know the area of ​​the parallelogram. Squares and rectangulars are parallelograms, like і be-like іnsha chotiristonnya figure, at which opposite sides are parallel. The area of ​​the parallelogram is calculated according to the following formula: S = bh, de "b" is the base (the lower side of the parallelogram), "h" is the height (from the upper to the lower side; the height is from the top to the lower side; the height is from the base to 90 °).

    • At the squares and rectangulars, the upper and lower sides and under the straight kut are hanging.

  1. Rip tricytes and parallelograms. Mіzh tsimi figures іsnu simple links. Like a parallelogram of diagonal razr_zati, viydut two equal tricycles. Similarly, if you put two tricytes on the side, you see a parallelogram. To that, the area of ​​any tricycle can be counted according to the following formula: S = ½bh to become half of the parallelogram area.

    Know the basis of the femoral tricycle. Now you know the formula for calculating the area of ​​the tricycle; overflowed with z'yasuvati, so also "pidstava" and "visota". The base (called yak "b") is the side that is not suitable for the two (equal) sides.

  2. Lower the perpendicular to the base. To grow tse from the tops of the tricot, yak is opposed to the base. Remember that the perpendicular flaps the base with a straight cut. Such a perpendicular is the visota of the tricycle (denoted as yak h). As soon as you know the value "h", you can calculate the area of ​​the tricycle.

    • In the femoral tricycle, the height of the base is exactly in the middle.

  3. Marvel at half a femoral tricycle. Beast to respect, that the height of the tricycle was divided into two equal tricytes. Marvel at one of them and know this side:

    • Short side of the back half of the base: b 2 (\ displaystyle (\ frac (b) (2))).
    • The other side is the price "h".
    • Hypotenuse of the rectangular tricycle є on the side of the femoral tricycle; meaning її yak "s".

  4. Hurry up on the Pythagorean theorem. Even though there are two sides of a rectangular tricycle, its third side can be calculated according to the Pythagorean theorem: (bik 1) 2 + (side 2) 2 = (hypotenuse) 2. In the application, Pythagoras' theorem is written as follows:.

    • Shvidshe for everything, Pythagoras' theorem is like this: a 2 + b 2 = c 2 (\ displaystyle a ^ (2) + b ^ (2) = c ^ (2))... The words "side 1", "side 2" and "hypotenuse" are alive and well;

  5. Calculate the value "h". Remember, that the formula for calculating the area of ​​the tricycle є changes "b" and "h", but the meaning of "h" is unavailable. Rewrite the formula, then calculate the "h":

    • (b 2) 2 + h 2 = s 2 (\ displaystyle ((\ frac (b) (2))) ^ (2) + h ^ (2) = s ^ (2))
      h 2 = s 2 - (b 2) 2 (\ displaystyle h ^ (2) = s ^ (2) - ((\ frac (b) (2))) ^ (2))
      .

  6. The formula is given in the form of the value and the number "h". The qiu formula can be stashed up to any kind of tricot, the side of which is seen. Replace "b" with the value of the base, and replace "s" with the value of the side, so that you know the value of "h".

    • At the butt: b = 6 div; s = 5 div.
    • Submit the following formula:
      h = (s 2 - (b 2) 2) (\ displaystyle h = (\ sqrt (()) s ^ (2) - ((\ frac (b) (2))) ^ (2))
      h = (5 2 - (6 2) 2) (\ displaystyle h = (\ sqrt (()) 5 ^ (2) - ((\ frac (6) (2))) ^ (2)))
      h = (25 - 3 2) (\ displaystyle h = (\ sqrt (()) 25-3 ^ (2)))
      h = (25 - 9) (\ displaystyle h = (\ sqrt (()) 25-9))
      h = (16) (\ displaystyle h = (\ sqrt (()) 16))
      h = 4 (\ displaystyle h = 4) div.

  7. Submit the value of the basis and the formula for calculating the area of ​​the tricycle. Formula: S = ½ bh; Submit in it the values ​​"b" and "h" and count the area. For example, do not forget to write a square unit for the video.

    • The base has a base of 6 divs, and a base of 4 divs.
    • S = ½bh
      S = ½ (6 cm) (4 cm)
      S = 12 cm2.

  8. Easy-to-read folding stock. In most cases, you will be given more important knowledge, you can see it in our application. It’s necessary to have a square root, which, as a rule, doesn’t fight nationally. In the first place, write down the value of the height at the viglyad of the common square root. Axle new stock:

    • Calculate the area of ​​the femoral tricycle, the sides of which are 8 cm, 8 cm, 4 cm.
    • In the base "b", vibrate the side, and the side is 4 cm.
    • Visota: h = 8 2 - (4 2) 2 (\ displaystyle h = (\ sqrt (8 ^ (2) - ((\ frac (4) (2))) ^ (2))))
      = 64 - 4 (\ displaystyle = (\ sqrt (64-4)))
      = 60 (\ displaystyle = (\ sqrt (60)))
    • Simplify the square root for an additional multiplier: h = 60 = 4 ∗ 15 = 4 15 = 2 15. (\ displaystyle h = (sqrt (60)) = (sqrt (4 * 15)) = (sqrt (4)) (sqrt (15)) = 2 (sqrt (15)).)
    • S = 1 2 b h (\ displaystyle = (\ frac (1) (2)) bh)
      = 1 2 (4) (2 15) (\ displaystyle = (\ frac (1) (2)) (4) (2 (\ sqrt (15)))))
      = 4 15 (\ displaystyle = 4 (\ sqrt (15)))
    • The view can be recorded with the root, or you can correct the root on the calculator and write down the view of the decimal fraction (S ≈ 15.49 cm 2).

Lіternі poznachennya sides and kutіv on the directed little unit will show the meanings, we mean in the formulas. With such a rank, it will help you to match the elements of the tricot tricycle. Take into account the details of the items you see, know on the armchairs the designated and accept the formula.

Formula of the area of ​​the femoral tricycle

Dali directed formulas for the knowledge of the area of ​​the femoral tricycle: through the sides, the bichu side that kut between them, through the bichnu side, the base that cot at the tops, through the bik base that cut when presented, etc. Just know the best way to go to the kid. For those who find the text right-handed, it will be explained why the formula є is correct and that it is possible to be found in the area.

  1. you can know know yogo bik that basis... Tsei viraz bulo otrimano by way of forgiveness of the larger, universal formulas. Just take Heron's formula as a basis, and then take the two sides of the tricycle with honey as a basis, and then say goodbye to the formula presented in the picture.
    The butt of the victorian of such a formula is aimed at the attachment of the problem solving below.
  2. Another formula allows you to know your area through bichnі sides and kut between them- half the square of the lateral sides, multiplied by the sine of the cut between the lateral sides
    If you think about lowering the hang on the side of the ryvno-femoral tricycle, it’s great, that її dovnina dorіvnuvatime a * sin β. Oscillations of the bite to the sides of the house to us, the visota is lowered on it now, half of the work and the large area of ​​this tricycle tricycle. є їх diagonals, which range іх equally navpіl, in such a rank, the area of ​​the femoral tricycle and the half of the side to the top). also Formula 5
  3. The third formula for showing the area across the side, the base is at the top.
    Strictly it seems, I know one of the grooves of a tricot tricycle, one can know that one of them is stuck in the forefront of the formula - nourishing relish (before a speech, one can remember only one of them).
    The third formula also has one tsikava specialty - tvir a sin α give us a whole lot of weight, lowered onto the base. As a result, the obvious simple and obvious formula 5.
  4. The area of ​​the femoral tricycle you can also know through bik basics that kut at delivery(kuti when presented rivni) as a square of the base, submissions on the chotiri tangential half of the kut, approved by both sides. Yakshto pride yourself respectfully, it becomes obvious that half of the base (b / 2) is multiplied by tg (β / 2) to give us the height of the tricot. Oscillations of the height in the tricot once bisectrix and median, then tg (β / 2) - the price of half of the base (b / 2) to the height - tg (β / 2) = (b / 2) / h. Stars h = b / (2 tan (β / 2)). In the result, the formula will be brought back to the simple Formula 5, which is quite obvious.
  5. Zrozumіlo, area of ​​the femoral tricycle You can know by lowering the height from the top to the base, as a result of which you will see two rectangular tricycles. Dali - everything is obvious. Half a twist on a baseі required area. Butt vikorystannya given formulas div. at tasks lower (2nd way of connection)
  6. The qia formula is to go, if you try to know the area of ​​the femoral tricycle behind the addition of Pyfagor's theorems... For a wide range of variations in front of the front formula, like one hour є with the leg of a rectangular tricycle, fixed by the side, half of the base and the head, through the Pyfagorian theorem. Bichna side є hypotenuse, to that from the square of the bichy side (s) the square of the other side is visible. Oskilki vіn dorіvnyu half of the base (b / 2) then the square is dorіvnyuvatime b 2/4. Forces of the root from tsyogo viraz and give us a hang. It can be seen in Formula 6. You should multiply the number and the standard by two, and then add the root sign for two numbers, we can accept another version of this formula, which is written through the sign "equal".
    Before the speech, you can play it, but if Formula 1 has the arches, then you will be reimagined to Formula 6. Anyway, the navpaki, the growth of squares of two numbers, is multiplied, to give us a wiggle, pershu.

Designation, like boules are stuck in the formulas for a little:

a- Dovzhina one from two sides of a trikutnik

b- Dovzhina bases

α - the size of one of two double kutіv at the podstavі

β - the size of the cut between the flat sides of the tricycle and the prototype yogic base

h- Dovzhina visoti, lowered from the top of the tricot tricycle to the base

Importantly... To brutalize respect for the signified wintry! Do not mix up α і β, and also aі b!

Note... Tse part of the lesson iz zavdannyi z geometrії (divided the area of ​​the tricot tricycle). There is a change of staff, as the victories are difficult for the hour of the date. If you need to update the geometrical problem, there is no way to write about it at the forum. For the designation of the square root, at the problem connections, the symbol √ or sqrt () is used, and at the arches it is meant to be viraz.

Zavdannya

The bichna side of the tricot tricycle is 13 cm long, and the base is 10 cm long. Know the area tricycle tricycle.



Decision.

1st way... Zastosuєmo formula of Heron. Oskilki trikutnik ravnoedreniy, then there is more than a simple viglyad (div. Formula 1 in the list of formulas vishche):

de a - dozhina of bichnyh parties, and b - dozhina of the basis.
Submitting the values ​​of the tricycle side from the mind, we can recognize:
S = 1/2 * 10 * √ ((13 + 5) (13 - 5)) = 5 √ (18 * 8) = 60 cm 2

2nd way. Zastosuєmo Pythagoras' theorem
It is permissible that the formula is remembered by Vikoristan as the first method of solution. It is lowered from the vertex B to the base AC to the height of the BK.
Oskilki of the height of the femoral tricycle, cut the base of the navpil, then half of the base will be equal
AK = AC / 2 = 10/2 = 5 cm.

Hanging with a half of the base and side of the tricot tricycle will fix the rectangular tricycle ABK. We have a hypotenuse AB and a leg AK for a tricycle. Virazimo adjusting the other leg through the Pyfagorian theorem.