Dovzhina arc. Kolo that of inscriptions kut. Visual guide (2019)


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A handful of items are saved on the site between the stake and the surrounding area. Schob poachit qiu rіznitsyu, enough to look, chim є offended fіguri. There is an unlimited number of points in the area, which are located at the same point from the central point. Ale, if there is a lot of storage from the inner space, then there is no way to lay it down. Walk around, where there is a colo and a colo, that is surrounded by that (o-circle), and an unlimited number of dots, that are in the middle of a cola.

For be-such a point L, how to lie on a number, equal to OL = R. (Dovzhina vіdrizka OL dorіvnyu radіusu cola).

Відрізок, which is from the two points of the stake, є її chordium.

Horda, go straight through the center of the stake, є diameter cola (D). The diameter can be calculated using the following formula: D = 2R

Dovzhina circle calculated by the formula: C = 2 \ pi R

Stake area: S = \ pi R ^ (2)

Arc stake to be called that її part, yaka roztashovutsya between two dots. Two points start two arcs of a stake. The chord CD tightens the two arcs: CMD and CLD. However, the jordi contract the same arcs.

Central kut to be called such a kut, which is located between two radii.

Dovzhinu arc you can know behind the formula:

  1. Vikoristovyuchi degree turn: CD = \ frac (\ pi R \ alpha ^ (\ circ)) (180 ^ (\ circ))
  2. Vikoristovuchi radialny zhіd: CD = \ alpha R

The diameter, which is perpendicular to the chord, extends the chord and the arcs are pulled together by it.

If the chordi AB and CD cola may peretin at the point N, then create the chords, split by the point N, equal to yourself.

AN \ cdot NB = CN \ cdot ND

Stosovno cola

Stosovno cola It is accepted to be called a straight line, for which є has one backward point from the colo.

There are two sleeping points near the straight line, they call sichucho.

I would like to draw a radius at the point of the torch, and it will be perpendicular to the point of the stake.

We will carry out two exact points to our stake. Look, where the dots grow one by one, and the center of the stake grows on the bisectric kuta with the top in the center of the point.

AC = CB

Now we will carry out the exact same way to the point from our point. Otrimaimo, that the square of the house is similar to the one that is the same as the one to the last part.

AC ^ (2) = CD \ cdot BC

You can add a new one: add a whole add-on of the first one to the first part of the day, add another add-on of another one to the last part.

AC \ cdot BC = EC \ cdot DC

Kuti in number

Degree of entering the central kut and arc, yaku to spiral, equal.

\ angle COD = \ cup CD = \ alpha ^ (\ circ)

Inscriptions kut- Tse kut, the top of which is to be found on the kolі, and the sides take revenge on the chordi.

It is possible to calculate it, having known the size of the arc, some fragments of the previous half of the arc.

\ angle AOB = 2 \ angle ADB

Yaky spiral on a diameter, inscribed kut, straight.

\ angle CBD = \ angle CED = \ angle CAD = 90 ^ (\ circ)

The inscribed kuti, which spiral on one arc, is the same.

Inscribed kuti based on the same chord, the same sum is available 180 ^ (\ circ).

\ angle ADB + \ angle AKB = 180 ^ (\ circ)

\ angle ADB = \ angle AEB = \ angle AFB

On one number there are the tops of the tricycles with the same kuts and the given basis.

The cut with the top of the middle of the stake and the seams between two chords is the same half of the sum of the core values ​​of the arcs of the stake, which polarize in the middle of the given vertical cut.

\ angle DMC = \ angle ADM + \ angle DAM = \ frac (1) (2) \ left (\ cup DmC + \ cup AlB \ right)

The cut with the top is the posture of the number of gussets between the two, the same half of the difference in the cut-off values ​​of the stake arcs, which are halfway between the middle of the cut.

\ angle M = \ angle CBD - \ angle ACB = \ frac (1) (2) \ left (\ cup DmC - \ cup AlB \ right)

Fitted colo

Fitted colo- Tse colo, why should the sides of the bagatokutnik be stuck.

At the point, the bisectors of the bagatokutnik are changing, the center is being developed.

Colo can be inscribed not in a leather bagatokutnik.

The area of ​​the bagatokutnik with the inscribed stake is behind the formula:

S = pr,

p - napіvperimeter bagatokutnik,

r is the radius of the inscribed stake.

Zvidsy viplya, the radius of the inscribed stake is delivered:

r = \ frac (S) (p)

Sumi dovzhin of the other sides will be the same, as the number is inscribed in the opukli chotirikutnik. I navpaki: in the opukli chotirikutnik, colo will fit in, as in the new sumi dozhin of the other sides to the same.

AB + DC = AD + BC

For any tricycle it is possible to enter a number. Leave one one. At the point where the bisectors of the inner cuts of the figuri are deformed, there is the center of the inscribed stake.

The radius of the inscribed stake is calculated according to the following formula:

r = \ frac (S) (p),

de p = \ frac (a + b + c) (2)

Described colo

If it is colo to pass through the skin top of the bagatokutnik, then it is taken to nazivati described by the bagatokutnik.

At the point of the overflow of the middle perpendiculars of the sides of the chain, the center of the described stake is rebuilt.

Radius can be known by counting the yak radius cola, yak is described by the tricutnik, which is designated as three peaks of the bagatokutnik.

Є taka umova: it is possible to describe closely the chotirikutnik deprived of it, as the sum of the other huts on the road 180 ^ (\ circ).

\ angle A + \ angle C = \ angle B + \ angle D = 180 (\ circ)

Bіlya whether a trikutnik can be described in a colo, moreover, only one. The center of such a stake will be seamed at the points, the middle perpendiculars of the sides of the tricycle are deconstructed.

The radius of the described stake can be calculated by the formulas:

R = \ frac (a) (2 \ sin A) = \ frac (b) (2 \ sin B) = \ frac (c) (2 \ sin C)

R = frac (abc) (4 S)

a, b, c - Dovzhini side tricutnik,

S - the area of ​​the tricycle.

Ptolemy's theorem

Nasamkinets, Ptolemy's theorem is understood.

Ptolemy's theorem is to deduce that there are more diagonals in the same sum of creations of the other sides of the inscribed chotirikutnik.

AC \ cdot BD = AB \ cdot CD + BC \ cdot AD

Colom I call a closed, flat curve, all the points that are near the same area, visible from the same points from the center.

Speck About є center stake, R є By the radius of the stake - I will go from the point of the stake to the center. For the value of a closed radio

Small. 1

curves may, however, be the same kind of dinner.

Vidstan between two stake points is called a chord. The tip of a stake, which passes through the center and the two points, is called a diameter. The middle of the diameter is the center of the stake. The stake points extend into a closed curve on two parts, the skin part is the arc of the stake. As soon as the end of the arc is to fit the diameter, then it is also called a pivkol, which is taken to mean π ... Degree turn is two years, which may turn into 360 degrees.

Concentric cola - the cola, which may be the main center. Orthogonal cola - a cola, which is re-swayed up to 90 degrees.

Ploshchina, the yak will be surrounded by a stake, called a stake. One part of a stake, a yak is surrounded by two radii and an arc - a whole circular sector. The arc of a sector is a chain arc that will surround the sector.

Small. 2

Instead, rostashuvannya stake that straight (Fig. 2).

The circumference and straight line are two sleeping points, which go straight ahead to the center of the mensha stake behind the radius of the stake. In such a way, it is straightforward to call it a cola.

The surroundings and straight lines go to one corner point, so that you go straight ahead to the center of the stake to the radius of the stake. At such a time, it is straightforward in terms of delivery to the stake to be called dota to the stake. ЇKhnya zagalna point to wear I’ll name the point dotsiku kola і straight.

Basic cola formulas:

  • C = 2πR , de C - more girth of the circumference
  • R = C / (2π) = D / 2 , de З / (2π) - Dovzhina arc stake
  • D = C / π = 2R , de D - Diameter
  • S = πR2 , de S - cola area
  • S = ((πR2) / 360) α , de S - The area of ​​the circular sector

Kolo and Kolo took their name from the Ancient Greece. Already in the old days, a man was tricked into a round body; Those that were so round could collapse by themselves, became a shipment to the wine-wheel of the wheel. Would you like to be so special? Ale to understand, as in one mile of a wheel to learn from our life. Nadal tsei vinakhid gave rise to the mathematical understanding of the stake.

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The formula for knowing how to finish the arc of a stake is simple, and even more often, on important DI-type experiences, such zavdannya is created, as it is ill-fated to see without storing. It is also necessary for the nobility to fold international standardized tests, for example, SAT and others.

Why should you buy a stake arc?

The wiggle's formula is as follows:

l = πrα / 180 °

She is a skin from the elements of the formula:

  • π is the number of Pi (a constant value, which is expensive ≈ 3.14);
  • r is the radius of the cola;
  • α - the size of the kuta, which spirals the arc (central, but not inscribed).

It can be seen that you can see the problem, in the mind you are guilty but that α. It is hard to know two values ​​without qih.

What is the formula and why are you looking at it?

Everything is even easier. Become zealous, as if put 360 ° on the banner, and put two on the day before. It is also possible α Do not overdo it in the fraction, just write it with a multiple sign. All in all, you can allow yourself, that the gift of the element will be worthy of the number. Todi zalny viglyad will become like this:

l = (2πr / 360 °) × α

Just for the sake of quickness, they speed up 2 and 360 °. And now, if you are amazed, then you can remember the formula of the dozen of all stake, but you can 2πr. Everything can be stored in 360 °, so there is a 360-degree fit. Let's multiply by a number α, Tobto for that bit of "shmatkiv pie", as we need. But all of it seems that the number (that is, the dinner of all the cola) cannot be extended by a degree. Well, how can such a person be robbed? Name, as a rule, the degree will speed up from the degree of the central kuta, tobto α. The idea is to become overwhelmed, and the result is to come up with a final result.

Tsim can explain to those for whom the arc of the stake is such a rank and such a viglyad.

The butt of the average folding due to the given formula

Umova: Є colo with a radius of 10 centimeters. Set the angle of the central cut to 90 °. To know a dozen of the arc of a stake, I am done with a tsim kut.

Solution: l = 10? × 90 ° / 180 ° = 10? × 1/2 = 5?

Type: l = 5π

So much as possible, the radial world of kuta was given to replace the degree approach. At the same time, it’s not easy to play, and even once it became easy. To convert the radial world into degrees, you need to multiply the number by 180 ° / π. Now, you can now submit a replacement α I will start a combination: m × 180 ° / π. De m - tse radianne meaning. A dal 180 and a number π To speed up and walk around, the formula is completely simplified, like a vigilade, like this:

  • m - radianna mira kuta;
  • r is the radius of the cola.