The formula for finding the length of an arc of a circle is quite simple, and very often in important exams such as the Unified State Exam there are problems that cannot be solved without its use. It is also necessary to know it to pass international standardized tests, such as SAT and others.

What is the length of the arc of a circle?

The formula looks like this:

l = πrα / 180°

What is each element of the formula:

• π - number Pi (constant value equal to ≈ 3.14);
• r is the radius of a given circle;
• α is the magnitude of the angle at which the arc rests (central, not inscribed).

As you can see, in order to solve the problem, r and α must be present in the condition. Without these two quantities, it is impossible to find the arc length.

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

Everything is extremely easy. It will become much clearer if you put 360° in the denominator and add two in the numerator. You can also α do not leave it in the fraction, take it out and write it with the multiplication sign. This is quite possible, since this element is in the numerator. Then the general view will be like this:

l = (2πr / 360°) × α

Just for convenience we shortened 2 and 360°. And now, if you look closely, you can see a very familiar formula for the length of the entire circle, namely - 2πr. The entire circle consists of 360°, so we divide the resulting measure into 360 parts. Then we multiply by the number α, that is, for the number of “pieces of the pie” that we need. But everyone knows for certain that a number (that is, the length of the entire circle) cannot be divided by a degree. What to do in this case? Usually, as a rule, the degree contracts with the degree of the central angle, that is, with α. After that, only numbers remain, and in the end the final answer is obtained.

This can explain why the length of the arc of a circle is found in this way and has this form.

An example of a problem of medium complexity using this formula

Condition: There is a circle with a radius of 10 centimeters. The degree measure of a central angle is 90°. Find the length of the circular arc formed by this angle.

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

Answer: l = 5π

It is also possible that instead of a degree measure, a radian angle measure would be given. Under no circumstances should you be afraid, because this time the task has become much easier. To convert a radian measure to a degree measure, you need to multiply this number by 180° / π. This means that now we can substitute α the following combination: m × 180° / π. Where m is the radian value. And then 180 and the number π are reduced and a completely simplified formula is obtained, which looks like this:

• m - radian measure of angle;
• r is the radius of a given circle.

How well do you remember all the names associated with the circle? Just in case, let us remind you - look at the pictures - refresh your knowledge.

Firstly - The center of a circle is a point from which the distances from all points on the circle are the same.

Secondly - radius - a line segment connecting the center and a point on the circle.

There are a lot of radii (as many as there are points on the circle), but All radii have the same length.

Sometimes for short radius they call it exactly length of the segment“the center is a point on the circle,” and not the segment itself.

And here's what happens if you connect two points on a circle? Also a segment?

So, this segment is called "chord".

Just as in the case of radius, diameter is often the length of a segment connecting two points on a circle and passing through the center. By the way, how are diameter and radius related? Look carefully. Of course, the radius is equal to half the diameter.

In addition to chords, there are also secants.

Remember the simplest thing?

Central angle is the angle between two radii.

And now - the inscribed angle

Inscribed angle - the angle between two chords that intersect at a point on a circle.

In this case, they say that the inscribed angle rests on an arc (or on a chord).

Look at the picture:

Measurements of arcs and angles.

Circumference. Arcs and angles are measured in degrees and radians. First, about degrees. There are no problems for angles - you need to learn how to measure the arc in degrees.

The degree measure (arc size) is the value (in degrees) of the corresponding central angle

What does the word “appropriate” mean here? Let's look carefully:

Do you see two arcs and two central angles? Well, a larger arc corresponds to a larger angle (and it’s okay that it’s larger), and a smaller arc corresponds to a smaller angle.

So, we agreed: the arc contains the same number of degrees as the corresponding central angle.

And now about the scary thing - about radians!

What kind of beast is this “radian”?

Imagine this: Radians are a way of measuring angles... in radii!

An angle of radians is a central angle whose arc length is equal to the radius of the circle.

Then the question arises - how many radians are there in a straight angle?

In other words: how many radii “fit” in half a circle? Or in another way: how many times is the length of half a circle greater than the radius?

Scientists asked this question back in Ancient Greece.

And so, after a long search, they discovered that the ratio of the circumference to the radius does not want to be expressed in “human” numbers like, etc.

And it’s not even possible to express this attitude through roots. That is, it turns out that it is impossible to say that half a circle is times or times larger than the radius! Can you imagine how amazing it was for people to discover this for the first time?! For the ratio of the length of half a circle to the radius, “normal” numbers were not enough. I had to enter a letter.

So, - this is a number expressing the ratio of the length of the semicircle to the radius.

Now we can answer the question: how many radians are there in a straight angle? It contains radians. Precisely because half the circle is times larger than the radius.

Ancient (and not so ancient) people throughout the centuries (!) tried to more accurately calculate this mysterious number, to better express it (at least approximately) through “ordinary” numbers. And now we are incredibly lazy - two signs after a busy day are enough for us, we are used to

Think about it, this means, for example, that the length of a circle with a radius of one is approximately equal, but this exact length is simply impossible to write down with a “human” number - you need a letter. And then this circumference will be equal. And of course, the circumference of the radius is equal.

Let's go back to radians.

We have already found out that a straight angle contains radians.

What we have:

So, glad, that is, glad. In the same way, a plate with the most popular angles is obtained.

The relationship between the values ​​of the inscribed and central angles.

There is an amazing fact:

The inscribed angle is half the size of the corresponding central angle.

Look how this statement looks in the picture. A “corresponding” central angle is one whose ends coincide with the ends of the inscribed angle, and the vertex is at the center. And at the same time, the “corresponding” central angle must “look” at the same chord () as the inscribed angle.

Why is this so? Let's look at a simple case first. Let one of the chords pass through the center. It happens like that sometimes, right?

What happens here? Let's consider. It is isosceles - after all, and - radii. So, (labeled them).

Now let's look at. This is the outer corner for! We recall that an external angle is equal to the sum of two internal angles not adjacent to it, and write:

That is! Unexpected effect. But there is also a central angle for the inscribed.

This means that for this case they proved that the central angle is twice the inscribed angle. But it’s a painfully special case: isn’t it true that the chord doesn’t always go straight through the center? But it’s okay, now this particular case will help us a lot. Look: second case: let the center lie inside.

Let's do this: draw the diameter. And then... we see two pictures that were already analyzed in the first case. Therefore we already have that

This means (in the drawing, a)

Well, that leaves the last case: the center is outside the corner.

We do the same thing: draw the diameter through the point. Everything is the same, but instead of a sum there is a difference.

That's all!

Let's now form two main and very important consequences from the statement that the inscribed angle is half the central angle.

Corollary 1

All inscribed angles based on one arc are equal to each other.

We illustrate:

There are countless inscribed angles based on the same arc (we have this arc), they may look completely different, but they all have the same central angle (), which means that all these inscribed angles are equal between themselves.

Corollary 2

The angle subtended by the diameter is a right angle.

Look: what angle is central to?

Certainly, . But he is equal! Well, therefore (as well as many more inscribed angles resting on) and is equal.

Angle between two chords and secants

But what if the angle we are interested in is NOT inscribed and NOT central, but, for example, like this:

or like this?

Is it possible to somehow express it through some central angles? It turns out that it is possible. Look: we are interested.

a) (as an external corner for). But - inscribed, rests on the arc -. - inscribed, rests on the arc - .

For beauty they say:

The angle between the chords is equal to half the sum of the angular values ​​of the arcs enclosed in this angle.

They write this for brevity, but of course, when using this formula you need to keep in mind the central angles

b) And now - “outside”! How to be? Yes, almost the same! Only now (again we apply the property of the external angle for). That is now.

And that means... Let’s bring beauty and brevity to the notes and wording:

The angle between the secants is equal to half the difference in the angular values ​​of the arcs enclosed in this angle.

Well, now you are armed with all the basic knowledge about angles related to a circle. Go ahead, take on the challenges!

CIRCLE AND INSINALED ANGLE. AVERAGE LEVEL

Even a five-year-old child knows what a circle is, right? Mathematicians, as always, have an abstruse definition on this subject, but we will not give it (see), but rather let us remember what the points, lines and angles associated with a circle are called.

Important Terms

Firstly:

center of the circle- a point from which all points on the circle are the same distance.

Secondly:

There is another accepted expression: “the chord contracts the arc.” Here in the figure, for example, the chord subtends the arc. And if a chord suddenly passes through the center, then it has a special name: “diameter”.

By the way, how are diameter and radius related? Look carefully. Of course,

And now - the names for the corners.

Natural, isn't it? The sides of the angle extend from the center - which means the angle is central.

This is where difficulties sometimes arise. Pay attention - NOT ANY angle inside a circle is inscribed, but only one whose vertex “sits” on the circle itself.

Let's see the difference in the pictures:

Another way they say:

There is one tricky point here. What is the “corresponding” or “own” central angle? Just an angle with the vertex at the center of the circle and the ends at the ends of the arc? Not certainly in that way. Look at the drawing.

One of them, however, doesn’t even look like a corner - it’s bigger. But a triangle cannot have more angles, but a circle may well! So: the smaller arc AB corresponds to a smaller angle (orange), and the larger arc corresponds to a larger one. Just like that, isn't it?

The relationship between the magnitudes of the inscribed and central angles

Remember this very important statement:

In textbooks they like to write this same fact like this:

Isn’t it true that the formulation is simpler with a central angle?

But still, let’s find a correspondence between the two formulations, and at the same time learn to find in the drawings the “corresponding” central angle and the arc on which the inscribed angle “rests”.

Look: here is a circle and an inscribed angle:

Where is its “corresponding” central angle?

Let's look again:

What is the rule?

But! In this case, it is important that the inscribed and central angles “look” at the arc from one side. For example:

Oddly enough, blue! Because the arc is long, longer than half the circle! So don’t ever get confused!

What consequence can be deduced from the “halfness” of the inscribed angle?

But, for example:

Angle subtended by diameter

Have you already noticed that mathematicians love to talk about the same thing in different words? Why do they need this? You see, the language of mathematics, although formal, is alive, and therefore, as in ordinary language, every time you want to say it in a way that is more convenient. Well, we have already seen what “an angle rests on an arc” means. And imagine, the same picture is called “an angle rests on a chord.” On what? Yes, of course, to the one that tightens this arc!

When is it more convenient to rely on a chord than on an arc?

Well, in particular, when this chord is a diameter.

There is a surprisingly simple, beautiful and useful statement for such a situation!

Look: here is the circle, the diameter and the angle that rests on it.

CIRCLE AND INSINALED ANGLE. BRIEFLY ABOUT THE MAIN THINGS

1. Basic concepts.

3. Measurements of arcs and angles.

An angle of radians is a central angle whose arc length is equal to the radius of the circle.

This is a number that expresses the ratio of the length of a semicircle to its radius.

The circumference of the radius is equal to.

4. The relationship between the values ​​of the inscribed and central angles.

Well, the topic is over. If you are reading these lines, it means you are very cool.

Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!

Now the most important thing.

You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough...

For what?

For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.

I won’t convince you of anything, I’ll just say one thing...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.

You won't be asked for theory during the exam.

You will need solve problems against time.

And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.

It's like in sports - you need to repeat it many times to win for sure.

Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!

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Find problems and solve them!

First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.

For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).

A segment that connects two points on a circle is its chord.

A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R

Circumference calculated by the formula: C=2\pi R

Area of ​​a circle: S=\pi R^(2)

Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.

Central angle An angle that lies between two radii is called.

Arc length can be found using the formula:

1. Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
2. Using radian measure: CD = \alpha R

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

If chords AB and CD of a circle intersect at point N, then the products of segments of chords separated by point N are equal to each other.

AN\cdot NB = CN\cdot ND

Tangent to a circle

Tangent to a circle It is customary to call a straight line that has one common point with a circle.

If a line has two common points, it is called secant.

If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.

Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.

AC = CB

Now let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.

AC\cdot BC = EC\cdot DC

Angles in a circle

The degree measures of the central angle and the arc on which it rests are equal.

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

Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.

You can calculate it by knowing the size of the arc, since it is equal to half of this arc.

\angle AOB = 2 \angle ADB

Based on a diameter, inscribed angle, right angle.

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

Inscribed angles that subtend the same arc are identical.

Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .

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

\angle ADB = \angle AEB = \angle AFB

On the same circle are the vertices of triangles with identical angles and a given base.

An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values ​​of the arcs of the circle that are contained within the given and vertical angles.

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

An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values ​​of the arcs of the circle that are contained inside the angle.

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

Inscribed circle

Inscribed circle is a circle tangent to the sides of a polygon.

At the point where the bisectors of the corners of a polygon intersect, its center is located.

A circle may not be inscribed in every polygon.

The area of ​​a polygon with an inscribed circle is found by the formula:

S = pr,

p is the semi-perimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is equal to:

r = \frac(S)(p)

The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.

AB + DC = AD + BC

It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors of the internal angles of the figure intersect, the center of this inscribed circle will lie.

The radius of the inscribed circle is calculated by the formula:

r = \frac(S)(p) ,

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

Circumcircle

If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.

At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.

The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.

There is the following condition: a circle can be described around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .

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

Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.

The radius of the circumscribed circle can be calculated using 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 are the lengths of the sides of the triangle,

S is the area of ​​the triangle.

Ptolemy's theorem

Finally, consider Ptolemy's theorem.

Ptolemy's theorem states that the product of diagonals is identical to the sum of the products of opposite sides of a cyclic quadrilateral.

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

Application

The length of the arc of a curved line in the Cartesian coordinate system on the site for students to consolidate the material they have covered. Calculating the length of the arc of a curve from an explicitly specified function using the application of an integral comes down to the basic calculation of a definite integral on a segment that is specified by the condition of the problem. Often you have to know how the length of the curve is calculated in practice without using available tools. This problem has been known to mathematicians like no other since ancient times, when a curve was straightened by dividing it into many straight segments, and their sum was calculated. At that time, great minds did not yet know that the value of the arc length can be calculated exactly only through the integral, while knowing its formula. It is the function of the argument that describes the line that is substituted into the formula and calculating the length of the curve after such manipulations is very, very simple. You can use our website calculator for this, specially designed for students and schoolchildren so that they do not waste a lot of time. There are many solvers on our resource, among which the length of the curve can be solved online in a matter of seconds and the result is displayed with high accuracy on the website page. In modern mathematics, each student needs to calculate the length of the arc of a curve as part of some specific problem, and perhaps along the way when performing more complex work. They even allocated a special lesson in which applications of a certain integral are studied using many formulas derived by scientists, among which there is such a task as finding the length of the arc of a curve from an explicitly or implicitly given function. the site is a calculator for calculating the arc length of a curve in a given coordinate system, which is studied in school and university curricula. In addition to the distance traveled according to the conditions of the text problem, the length of the arc of the curve is also calculated through the integral, precisely the one that corresponds to the trajectory of the pedestrian’s movement. Without touching on abstract objects or ordinary complex calculus systems, for example, such as Riemannian space, which affect affine transformations, we will assume that the length of the arc of the curve is in the Cartesian coordinate system. Therefore, feel free to visit the site, where a section on finding the length of an arc online is presented. Generally speaking, the value of the length of the curve in different coordinate systems will also be different and this is an indisputable fact, but very interesting. Let's say we have a curvilinear coordinate system in three-dimensional space, and the length of the curve depends on the starting and ending points of the segment under study. So, if you display a curve point by point in this system, then visually it will represent a straight line with respect to rectangular coordinates and the length of the arc will be determined through the usual integral. But it is sometimes very difficult to come to this non-obvious fact experimentally, and by nature, humans are not able to visually imagine a curved space. We reduce everything to comparison with the familiar rectangular Cartesian system. However, the length of a curve can be calculated in any coordinate system if it is rectifiable, which is the basis for solving such a problem. There are non-rectifiable curves, the length of the curve of which no calculator can find online. In general, such a curve cannot be specified in normal representation. There is a rule by which it is built and that’s it. But no one will be able to calculate the length of the arc of a curve, because it simply does not exist. In addition to calculating the mass of a body or the moments of inertia of bodies and plates, by applying a definite integral, you can easily find the arc length of a curved line in various coordinate systems, even in polar, by the way, this is even easier to do than in Cartesian. The transition from the first coordinate system to the second is performed by a simple transformation through trigonometric functions, determining the relationship between the angle and radius function from the origin. Next, you can use the free curve arc length calculator and get the result right here in the browser window. The website will be able to complete the website instantly and will help in solving the task. In this case, the arc length will be found online from any function from any higher mathematics problem book. In the course of the mathematical solution, the length of the curve can be found through a definite integral of a given function over the lower and upper limits of integration. Simultaneously with finding a general solution to the problem, the length of the arc through a specific integral is determined instantly by substituting the required quantities into the final sum of the integrand, which leads us to the study of the radical function at the ends of the segment. At the same time, a parallel one is added to the mathematical problem, namely, to correctly calculate the length of a curved line using the variable t, as a function of the time scale along the direction of movement of the material point. However, the trajectory of movement of a body as a collection of material points, or a specific point separately, cannot represent the full nature of the movement without having a law by which its line of movement is described. But you can, of course, using mathematical analysis, easily study its movement, including acceleration in certain areas, and also determine what the length of the curve is and calculate this value online on the website. How to find the arc length of a curve through an integral has been known for a long time to scientists all over the world, but conveying this to students in a timely manner is not as easy as it seems at first glance. This wonderful and useful resource for students and schoolchildren allows you to calculate the length of the arc of a curve, as they say, here and now right before your eyes, and the answer will be absolutely error-free to the nearest thousandths. Since applications of a definite integral are not fully studied in universities, because little study time is allocated for this, students themselves need to make an effort in understanding this important branch of mathematics, because it will be useful later in life. Similar sites, unlike the site, will also help you find the length of the arc of a curve from a given function, but we still recommend comparing the computational results with the answer received from our resource, for which we stand behind our reputation. No matter what modern and powerful arc length calculator you use to study a problem in mathematics, not a single such calculator or computer can help you learn how to solve such problems yourself. You are required to be attentive, persevering, and most importantly, patient when studying any mathematical section, since this is an exact science and requires gradual mastery due to its specifics. Returning to our topic, let’s summarize our thoughts on how to find out without much difficulty that the arc length of a curve can be found using a predetermined formula and calculated correctly. We strongly advise everyone who begins to study the science of mathematics to be able to and know how the length of an arc is calculated online on the website. As usual, in such problems, such a length of the arc of the curve, calculated through the integral, is necessary, since the course of further solving the problem depends on this. The purpose of the condition for identifying critical points is not directly related to the task at hand, but the length of the curve is determined according to the same principles of mathematical laws. The students correctly noted that the arc length as a definite integral gives the maximum answers to all the questions posed when studying the area of ​​motion of a material point. When studying the movement of a material plate, it is enough to know the trajectory of movement of at least two of its points, because the lines of all its other points can be determined based on geometric relationships, especially then the length of the movement curve of any point on the plate can be calculated. Using the calculator on the site, the length of the curve is determined online almost immediately and with the highest accuracy in the answer, since we use modern technologies in the approach to solving such mathematical problems. 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Your friends and colleagues will not be able to identify faster than you that the arc length of the curve will be greater for the first function, but not for the second function, although the segments are taken to be the same from the first point to the last. It happens that the length of the arc online has a value different from that obtained by manual calculation using available calculators and tables, but you should not draw unfounded conclusions ahead of time, because natural error is permissible in manual calculations. In conclusion, students still need to recommend the site site and clearly say that the length of the arc of a curve through the integral is calculated much faster, more productively and much more accurately than using numerical methods for approximate calculations. It is widely accepted at the student level that the length of a curve cannot be calculated through mathematical formulas unless one knows them. However, no, this is an erroneous judgment, because there are modern services on the Internet, specially tailored for students, where there are calculators for mathematics and the arc length as an integral is calculated in a matter of seconds right before the eyes of site users. Such calculators can calculate the length of the curve from any given function, using mathematical laws to do this, identifying and selecting optimal calculations, just as a competent teacher would do. After all, programming an arbitrarily complex computational process is more profitable and less expensive than constantly delving into abstruse books, looking for a suitable formula, determining the type of function, and so on and so forth. 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As soon as the teacher explains a theory, he immediately gives an example to reinforce the material covered. So the length of the curve can be found using numerical methods or using an integral, which is more preferable. Because the arc length as an integral of a function can be calculated in different ways, but this exactly gives the desired result and it can be further used in calculations of any type of problem. Since calculating the length of the curve is required in almost every lesson, while the teacher delivers the material as it becomes more complex, we advise you to choose our website called website and make your work easier. because that's what young people do today. They don’t get hung up on some complicated example, but as soon as they encounter obstacles on their way, they take it head on and solve problems using a calculator. 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However, no matter how good the numerical method is to find the length of the arc of the curve, but through the integral this length will still have an exact value, so to speak, a reference value in comparison with essentially similar numerical results.

Initially it looks like this:

Figure 463.1. a) existing arc, b) determination of segment chord length and height.

Thus, when there is an arc, we can connect its ends and get a chord of length L. In the middle of the chord we can draw a line perpendicular to the chord and thus get the height of the segment H. Now, knowing the length of the chord and the height of the segment, we can first determine the central angle α, i.e. the angle between the radii drawn from the beginning and end of the segment (not shown in Figure 463.1), and then the radius of the circle.

The solution to such a problem was discussed in some detail in the article “Calculation of an arched lintel”, so here I will only give 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 you can see, from a mathematical point of view, there are no problems with determining the radius of a circle. This method allows you to determine the value of the arc radius with any possible accuracy. This is the main advantage of this method.

Now let's talk about the disadvantages.

The problem with this method is not even that you need to remember formulas from a school geometry course, successfully forgotten many years ago - in order to recall the formulas - there is the Internet. And here is a calculator with functions arctg, arcsin, etc. Not every user has it. And although this problem can also be successfully solved by the Internet, we should not forget that we are solving a fairly applied problem. Those. It is not always necessary to determine the radius of a circle with an accuracy of 0.0001 mm; an accuracy of 1 mm may be quite acceptable.

In addition, in order to find the center of the circle, you need to extend the height of the segment and plot a distance on this straight line equal to the radius. Since in practice we are dealing with non-ideal measuring instruments, we should add to this the possible error in marking, it turns out that the smaller the height of the segment in relation to the length of the chord, the greater the error may occur when determining the center of the arc.

Again, we should not forget that we are not considering an ideal case, i.e. This is what we immediately called the curve an arc. In reality, this may be a curve described by a rather complex mathematical relationship. Therefore, the radius and center of the circle found in this way may not coincide with the actual center.

In this regard, I want to offer another method for determining the radius of a circle, which I often use myself, because this method of determining the radius of a circle is much faster and easier, although the accuracy is much less.

Second method for determining the radius of the arc (method of successive approximations)

So let's continue to consider the current situation.

Since we still need to find the center of the circle, to begin with, we will draw at least two arcs of arbitrary radius from the points corresponding to the beginning and end of the arc. Through the intersection of these arcs there will be a straight line, on which the center of the desired circle is located.

Now you need to connect the intersection of the arcs with the middle of the chord. However, if we draw not one arc from the indicated points, but two, then this straight line will pass through the intersection of these arcs and then it is not at all necessary to look for the middle of the chord.

If the distance from the intersection of the arcs to the beginning or end of the arc in question is greater than the distance from the intersection of the arcs to the point corresponding to the height of the segment, then the center of the arc in question is located lower on the straight line drawn through the intersection of the arcs and the midpoint of the chord. If it is less, then the desired center of the arc is higher on the straight line.

Based on this, the next point on the straight line is taken, presumably corresponding to the center of the arc, and the same measurements are made from it. Then the next point is accepted and the measurements are repeated. With each new point, the difference in measurements will become less and less.

That's all. Despite such a lengthy and complicated description, 1-2 minutes are enough to determine the radius of the arc in this way with an accuracy of 1 mm.

In theory it looks something like this:

Figure 463.2. Determination of the center of the arc by the method of successive approximations.

But in practice it goes something like this:

Photo 463.1. Marking workpieces of complex shapes with different radii.

Here I’ll just add that sometimes you have to find and draw several radii, because there’s so much mixed up in the photograph.