Rozrakhunok dovzhini arc in radius and direction. Kolo ta inscriptions kut. Visual guide (2019)

Kolom is called a closed, flat curve, all points of which are on the same plane, but are removed from the center.

Krapka About є the center of the stake, R With the radius of the stake - I will stand from any point of the stake to the center. Beyond the indicated radii, the radius is closed

Small 1

However, the crooked ones are looming.

Standing between two stake points is called a chord. The length of the stake that passes through its center and connects the two points is called the diameter. The middle of the diameter is the center of the stake. The stake points divide the closed curve into two parts, the skin part forms the arc of the stake. If the ends of the arc lie on the diameter, then it is called a circle, the end of which is usually designated π . The degree angle of the two coils, which swing the joint ends, becomes 360 degrees.

Concentric cola - the same cola that marks the celestial center. Orthogonal stakes are circles that move at 90 degrees.

The area that encloses the stake is called a stake. One part of the stake is surrounded by two radii and an arc - this is a circular sector. A sector arc is an arc that encloses a sector.

Small 2

Mutually stir the stake and straight (Fig. 2).

The circle and the straight line mark two points of contact, so that they stand from the straight line to the center of the stake less than the radius of the stake. In this case, directly in relation to the stake, it is called juicy.

The circle and the straight lines mark one corner point, so that the straight line reaches the center of the stake, which is equal to the radius of the stake. In this case, directly in relation to the stake, it is called a fraction to the stake. This point is called the dotika kola point and the straight line.

Basic formulas of cola:

C = 2πR , de C - dovzhina cola
R = С/(2π) = D/2 , de W/(2π) - Dovzhina arc stake
D = C/π = 2R , de D - Diameter
S = πR2 , de S - stake area
S = ((πR2)/360)α , de S - Area of the circular sector

Kolo and Kolo took their name from Ancient Greece. Even in the old days, people were marked with round bodies, which was the end of thoroughness. Those whose round bodies could collapse on their own became a mess until the wheel exited. I wonder what is special about this wine? Let us realize that in one stroke the wheels will disappear from our life. Next, this vinakhid gave rise to the mathematical concept of cola.

The formula for finding a long-term stake arc is simple, and even often in important tests of the EDI type, such problems arise that it is impossible to solve without stagnation. It is also necessary to know this for the development of international standardized tests, for example, SAT and others.

Why is the anniversary of the arc of the stake dear?

The formula looks like this:

l = πrα / 180°

What is skin based on the elements of the formula:

π - number Pi (stationary value, which is more than ≈ 3.14);
r - radius of the stake;
α is the size of the cut that the arc spirals (central, and not inscriptions).

Apparently, in order to complete the task, it is in mind that α is to blame. Without two values, it is impossible to know the dovzhin of the arc.

How is this formula derived and why does it look like this?

Everything is so easy. You will become more intelligent if you put 360° in the sign, and add a two in front of the number. It's also possible α Do not leave out the fraction, write її and write і with a multiplication sign. You can completely afford it, since this element is with the number manager. Then you will look like this:

l = (2πr / 360°) × α

Just for clarity, we did 2 and 360°. And now, if you are surprised, you can note the already known formula for doubling every stake, and yourself - 2πr. The whole circle is folded into 360°, so we divide the approach into 360 parts. Then we multiply by the number α, That's enough for the amount of "pie pieces" we need. But everyone knows that the number (the day of every stake) cannot be divided by a degree. Why work for such a time? As a rule, the degree is shortened by the degree of the central corner, so α. Then they are deprived of numbers, and the results come out in the end.

This can be explained by why the dove of the arc of the stake is known in such a manner and has such a look.

The butt is of medium foldability based on the formulation of this formula.

Umova: With a radius of 10 centimeters. The degree angle of the central bank is set to 90°. Know the dovzhina of the arc of the stake, established by tzim kut.

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

Version: l = 5π

It is also possible that instead of the degree setting, the radial world would be given. Every once in a while it’s not easy to talk, and even then the problem has become much easier. To convert the radial world into degrees, you need to multiply this number by 180 ° / π. Hey, now you can substitute α Combination: m × 180 ° / π. De m is not radiant. And then 180 is the number π to hurry up and leave is a simplified formula, as it looks like the coming order:

m - radian of the Kuta world;
r is the radius of the stake.

Right off the bat it looks like this:

Malyunok 463.1. a) an arc is visible; b) vyznachennya dovzhini khordi segmenta ta visoti.

In this way, if there is an arc, we can connect its ends and take off the chord of the chord L. In the middle of the chord we can draw a line perpendicular to the chord and in this way we take off the height of the segment H. Now, knowing the depth of the chord and the height of the segment, We can start now significant central cut α, tobto. where between the radii drawn from the cob to the end of the segment (not shown on the baby 463.1), and then the radius of the stake.

The essence of such a plant was clearly seen in the article “The Rose of the Arched Bridge,” so here I’ll give you the basic formulas:

tg( a/4) = 2N/L (278.1.2)

A/4 = arctan( 2H/L)

R = H/(1 - cos( a/2)) (278.1.3)

As a matter of fact, according to mathematics, there are no everyday problems related to the values of the radius of the stake. This method allows you to calculate the arc radius with any accuracy. The main advantage of this method.

Now let's talk about the shortcomings.

The problem with this method is not that it is necessary to memorize formulas from a school geometry course; it is successfully forgotten in many ways - in order to guess the formulas - but on the Internet. And the axis calculator has the function arctg, arcsin and other. є far from being a skin problem. And although this problem can also be successfully solved on the Internet, we must not forget that we can complete our practical tasks. Tobto. It will soon be necessary to calculate the radius of the stake with an accuracy of up to 0.0001 mm; an accuracy of 1 mm may be quite acceptable.

In addition, in order to find the center of the stake, it is necessary to extend the height of the segment and the insertion on this straight line, which is the same as the radius. So, in practice, we can deal with non-ideal vibrating adjustments, to which extent we can add a possible error in the distribution, it turns out that the lower the height of the segment in relation to the last chord, the more we can achieve the error in the assigned to the center of the arc.

Once again, there is a trace of forgetting in what we see as not a perfect fit, then. We immediately called the crooked arc. In reality, there may be a curve, which is described by a folding mathematical formula. And thus, the radius and center of the stake may not coincide with the actual center.

In connection with this, I want to introduce another method for calculating the radius of a stake, which I myself often use, because this method of calculating the radius of a stake is much faster and simpler, although the accuracy is significantly less.

Another method for determining the radius of the arc (method of successive approaches)

So, let’s continue to look at the obvious situation.

Since we still need to know the center of the stake, then for the cob from the point that indicates the cob and the end of the arc, we will draw at least two arcs of sufficient radius. A straight line will pass through the crossbar of these arcs, which will be the center of the stake.

Now you need to connect the crossbar of the arches from the middle of the chord. Moreover, since from the designated points we draw not one arc, but two, then we pass straight through the webbing of these arcs and thus searching the middle of the chord is not at all obligatory.

If you stand from the crossbar of the arcs to the beginning or end of the arc, which is seen more, then stand from the crossbar of the arcs to a point corresponding to the height of the segment, which means the center of the arc is located lower on the straight line, passing through the crossbar of the arcs and the middle of the chord. If less, then the search for the center of the arc is on the straight line.

Therefore, a point is taken on a straight line, which is quite similar to the center of the arc, and the same vibrations vibrate from it. Then the point is reached and the events are repeated. With a new skin point, the diversity of the worlds will be less.

Axis, power, and that’s it. Do not be fooled by simple and folded tabletops, to determine the arc radius in this way, with an accuracy of up to 1 mm, 1-2 pins are sufficient.

Theoretically, it looks something like this:

Malyunok 463.2. Designated to the center of the arc by the path of the successive ones closer.

But in practice it’s something like this:

Svitlina 463.1. Layout of a folding workpiece with different radii.

Here I’ll just add that sometimes you happen to find a lot of radii in the chair, so there’s so much stuff in the photograph.

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Now let’s look at the difference between the stake and the circle. To understand this difference, it is enough to take a look at what is wrong with the figures. There is an indisputable number of points on the plane that are located on a level surface from a single central point. Well, if the circle is formed from the internal space, then the circle will not fit. Find out that there is a circle and a circle that encloses the circle, and an untreated number of points in the middle of the stake.

For any point L that lies on the stake, the equation is OL=R. (Double cut OL is similar to the radius of the stake).

A cut that connects two points of the stake, and chordia.

The chord that goes straight through the center of the stake is diameter whose stake (D) . The diameter can be calculated using the following formula: D=2R

Dovzhina stake calculated using the formula: C=2\pi R

Square stake: S=\pi R^(2)

Dugogo stake This part is called as it is divided between two points. These two points indicate two arcs of the stake. The CD chord connects two arcs: CMD and CLD. However, new chords tie together new arches.

Central Kut This is called a zone that lies between two radii.

Dovzhinu arc can be found by the formula:

Vikorist and degree setting: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
Vikorist radial approach: CD = \alpha R

The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it.

If the AB and CD chords draw a crossbar at point N, then create split chords, separated by point N, that are equal to each other.

AN\cdot NB = CN\cdot ND

Stosovno cola

Stosovno cola It is customary to call it straight, which has one point behind the colo.

Since a straight line has two points, they are called sichuchoi.

If you want to draw a radius at the torcaning point, it will be perpendicular to the point of the stake.

Let's draw two additional points from this point to our stake. You will see that the sections of the subordinates are aligned one by one, and the center of the stake extends along a bisector between the vertex and the vertex at this point.

AC = CB

Now, to the stake from our point, let’s carry out the research and sichnu. It is important to note that the square of the final cut will ensure that all the cuttings are added to the outer part.

AC^(2) = CD \cdot BC

You can make a substitution: adding a whole piece of first juice to your outer part and adding a whole piece of another juice to your outer part.

AC\cdot BC = EC\cdot DC

Kuti in koli

Degrees of the central ridge and the arc in which it spirals, the plains.

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

Inscriptions kut- Tse kut, the top of which is on the stake, and the sides are on the chords.

You can calculate it by finding out the magnitude of the arc, which is still half the value of the arc.

\angle AOB = 2 \angle ADB

Which spirals to a diameter, fits cut, straight.

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

The inscribed lines that spiral onto one arc are the same.

The inscriptions that rest on the same chord are the same as the sum equal to 180^ (\circ) .

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

\angle ADB = \angle AEB = \angle AFB

On one side there are the tops of the tricutniks with the same tunics and a given base.

With the apex in the middle of the stake and the extensions between the two chords, the same half is the sum of the core values of the stake arcs, which lie in the middle of this vertical cut.

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

At the top of the pole there is a spread between the two sides, and the same half of the difference in the size of the arcs of the stake, which lie in the middle of the pile.

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

Inscribed colo

Inscribed colo- This is what is happening on the rich side.

At the point where the bisectors of the ornate bushes intertwine, its center expands.

Kolo mozhe buti is inscribed not in the leather bagatokutnik.

The area of the orchard with the inscribed stake is determined by the formula:

S = pr,

p is the perimeter of the rich-cothedule,

r is the radius of the inscribed stake.

The star indicates that the radius of the inscribed stake is equal to:

r = \frac(S)(p)

The sums of the two sides will be the same as those inscribed on the bulging chotirikutnik. And just a note: the rounded chotirikutnik fits into the colo, since this sum has two sides of the same.

AB + DC = AD + BC

If you have trikutniks, you can write a colo. Just one more. At the point where the bisectors of the figure’s inner corners intertwine, the center of its inscribed stake lies.

The radius of the inscribed stake is calculated using the following formula:

r = \frac(S)(p) ,

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

Described colo

If a colo passes through the cutaneous apex of the origami, then the colo is commonly called described the white of the algae cucumber.

At the point of the crossbar of the middle perpendiculars of the sides of this figure will be the center of the described stake.

The radius can be known by calculating it as the radius of the stake, which is described by the tricutaneous body, designated by the three apexes of the rich cuticle.

And this is how I think: it’s impossible to describe anything close to the chotirikutnik, since the sum of its long-lasting kuti is as high as 180^(\circ) .

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

You can describe the entire body of any trikutnik, and one at a time. The center of such a stake will be spread out at the point where the middle perpendiculars of the sides of the tricut will overlap.