The part of a figure that forms a circle whose points are equidistant is called an arc. If we draw rays from the center point of the circle to points coinciding with the ends of the arc, its central angle will be formed.

Determining arc length

Produced according to the following formula:

where L is the desired arc length, π = 3.14, r is the radius of the circle, α is the central angle.

The length of the arc of a circle is 14.82 centimeters.

In elementary geometry, an arc is understood as a subset of a circle located between two points located on it. In practice, solve problems in definition her length engineers and architects have to do it quite often, since this geometric element is widespread in a wide variety of designs.

Perhaps the first to face this task were the ancient architects, who in one way or another had to determine this parameter for the construction of vaults, widely used to cover the gaps between supports in round, polygonal or elliptical buildings. If you take a close look at the masterpieces of ancient Greek, ancient Roman and especially Arab architecture that have survived to this day, you will notice that arches and vaults are extremely common in their designs. The creations of modern architects are not so rich in them, but these geometric elements are, of course, present in them.

Length various arc must be calculated during the construction of roads and railways, as well as motor tracks, and in many cases traffic safety largely depends on the correctness and accuracy of the calculations. The fact is that many turns of highways, from a geometric point of view, are precisely arcs, and as they move along them, various physical forces act on vehicles. The parameters of their resultant are largely determined by the length of the arc, as well as its central angle and radius.

Designers of machines and mechanisms have to calculate the lengths of various arcs for the correct and accurate arrangement of the components of various units. In this case, errors in calculations are fraught with the fact that important and critical parts will interact incorrectly with each other and the mechanism simply will not be able to function as its creators plan. Examples of structures that are replete with geometric elements such as arcs include internal combustion engines, gearboxes, wood and metalworking equipment, body parts of cars and trucks, etc.

Arcs They are quite common in medicine, particularly in dentistry. For example, they are used to correct malocclusions. Corrective elements called braces (or bracket systems) and having the appropriate shape, are made of special alloys, and are installed in such a way as to change the position of the teeth. It goes without saying that in order for the treatment to be successful, these arcs must be very accurately calculated. In addition, arches are very widely used in traumatology, and perhaps the most striking example of this is the famous Ilizarov apparatus, invented by a Russian doctor in 1951 and extremely successfully used to this day. Its integral parts are metal arcs, equipped with holes through which special knitting needles are threaded, and which are the main supports of the entire structure.

Problems on finding the area of ​​a circle are a mandatory part of the Unified State Examination in mathematics. As a rule, several tasks are assigned to this topic in the certification test. All high school students, regardless of their level of preparation, should understand the algorithm for finding the circumference and area of ​​a circle.

If such planimetric tasks cause you difficulties, we recommend that you turn to the Shkolkovo educational portal. With us you can fill gaps in knowledge.

The corresponding section of the site presents a large selection of problems for finding the circumference and area of ​​a circle, similar to those included in the Unified State Exam. Having learned to perform them correctly, the graduate will be able to successfully cope with the exam.

Basic moments

Problems that require the use of area formulas can be direct or inverse. In the first case, the parameters of the figure elements are known. In this case, the required quantity is area. In the second case, on the contrary, the area is known, and it is necessary to find some element of the figure. The algorithm for calculating the correct answer in such tasks differs only in the order in which the basic formulas are applied. That is why, when starting to solve such problems, it is necessary to repeat the theoretical material.

The educational portal “Shkolkovo” provides all the basic information on the topic “Finding the length of a circle or arc and the area of ​​a circle,” as well as on other topics, for example. Our specialists prepared it and presented it in the most accessible form.

Having remembered the basic formulas, students can begin to complete problems for finding the area of ​​a circle, similar to those included in the Unified State Exam, online. For each exercise, the site provides a detailed solution and the correct answer. If necessary, any task can be saved in the “Favorites” section in order to return to it later and discuss it with the teacher.

The circle, its parts, their sizes and relationships are things that a jeweler constantly encounters. Rings, bracelets, castes, tubes, balls, spirals - a lot of round things have to be made. How can you calculate all this, especially if you were lucky enough to skip geometry classes at school?..

Let's first look at what parts a circle has and what they are called.

• A circle is a line that encloses a circle.
• An arc is a part of a circle.
• Radius is a segment connecting the center of a circle with any point on the circle.
• A chord is a segment connecting two points on a circle.
• A segment is a part of a circle bounded by a chord and an arc.
• A sector is a part of a circle bounded by two radii and an arc.

The quantities we are interested in and their designations:

Now let's see what problems related to parts of a circle have to be solved.

• Find the length of the development of any part of the ring (bracelet). Given the diameter and chord (option: diameter and central angle), find the length of the arc.
• There is a drawing on a plane, you need to find out its size in projection after bending it into an arc. Given the arc length and diameter, find the chord length.
• Find out the height of the part obtained by bending a flat workpiece into an arc. Source data options: arc length and diameter, arc length and chord; find the height of the segment.

Life will give you other examples, but I gave these only to show the need to set some two parameters to find all the others. This is what we will do. Namely, we will take five parameters of the segment: D, L, X, φ and H. Then, choosing all possible pairs from them, we will consider them as initial data and find all the rest by brainstorming.

In order not to unnecessarily burden the reader, I will not give detailed solutions, but will present only the results in the form of formulas (those cases where there is no formal solution, I will discuss along the way).

And one more note: about units of measurement. All quantities, except the central angle, are measured in the same abstract units. This means that if, for example, you specify one value in millimeters, then the other does not need to be specified in centimeters, and the resulting values ​​will be measured in the same millimeters (and areas in square millimeters). The same can be said for inches, feet and nautical miles.

And only the central angle in all cases is measured in degrees and nothing else. Because, as a rule of thumb, people who design something round don't tend to measure angles in radians. The phrase “angle pi by four” confuses many, while “angle forty-five degrees” is understandable to everyone, since it is only five degrees higher than normal. However, in all formulas there will be one more angle - α - present as an intermediate value. In meaning, this is half the central angle, measured in radians, but you can safely not delve into this meaning.

1. Given the diameter D and arc length L

; chord length ;
segment height ; central angle .

2. Given diameter D and chord length X

; arc length ;
segment height ; central angle .

Since the chord divides the circle into two segments, this problem has not one, but two solutions. To get the second, you need to replace the angle α in the above formulas with the angle .

3. Given the diameter D and central angle φ

; arc length ;
chord length ; segment height .

4. Given the diameter D and height of the segment H

; arc length ;
chord length ; central angle .

6. Given arc length L and central angle φ

; diameter ;
chord length ; segment height .

8. Given the chord length X and the central angle φ

; arc length ;
diameter ; segment height .

9. Given the length of the chord X and the height of the segment H

; arc length ;
diameter ; central angle .

10. Given the central angle φ and the height of the segment H

; diameter ;
arc length ; chord length .

The attentive reader could not help but notice that I missed two options:

5. Given arc length L and chord length X

7. Given the length of the arc L and the height of the segment H

These are just those two unpleasant cases when the problem does not have a solution that could be written in the form of a formula. And the task is not so rare. For example, you have a flat piece of length L, and you want to bend it so that its length becomes X (or its height becomes H). What diameter should I take the mandrel (crossbar)?

This problem comes down to solving the equations:
; - in option 5
; - in option 7
and although they cannot be solved analytically, they can be easily solved programmatically. And I even know where to get such a program: on this very site, under the name . She does everything that I’m telling you at length here in microseconds.

To complete the picture, let’s add to the results of our calculations the circumference and three area values ​​- circle, sector and segment. (Areas will help us a lot when calculating the mass of all round and semicircular parts, but more on this in a separate article.) All these quantities are calculated using the same formulas:

circumference ;
area of ​​a circle ;
sector area ;
segment area ;

And in conclusion, let me remind you once again about the existence of an absolutely free program that performs all of the above calculations, freeing you from the need to remember what an arctangent is and where to look for it.

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Circumference called a closed, plane curve, all points of which, lying in the same plane, are located at the same distance from the center.

Dot ABOUT is the center of the circle, R is the radius of the circle - the distance from any point on the circle to the center. By definition, all radii of a closed

rice. 1

the curves have the same length.

The distance between two points on a circle is called a chord. A segment of a circle passing through its center and connecting two of its points is called a diameter. The midpoint of the diameter is the center of the circle. Points on a circle divide a closed curve into two parts, each part is called a circular arc. If the ends of the arc belong to the diameter, then such a circle is called a semicircle, the length of which is usually denoted π . The degree measure of two circles that have common ends is 360 degrees.

Concentric circles are circles that have a common center. Orthogonal circles are circles that intersect at an angle of 90 degrees.

The plane enclosed by a circle is called a circle. One part of the circle, which is limited by two radii and an arc, is a circular sector. A sector arc is an arc that bounds a sector.

Rice. 2

The relative position of a circle and a straight line (Fig. 2).

A circle and a straight line have two points in common if the distance from the straight line to the center of the circle is less than the radius of the circle. In this case, the straight line in relation to the circle is called a secant.

A circle and a straight line have one common point if the distance from the straight line to the center of the circle is equal to the radius of the circle. In this case, the line in relation to the circle is called tangent to the circle. Their common point is called the tangency point of the circle and the line.

Basic circle formulas:

• C = 2πR , Where C - circumference
• R = С/(2π) = D/2 , Where С/(2π) — length of the arc of a circle
• D = C/π = 2R , Where D - diameter
• S = πR2 , Where S - area of ​​a circle
• S = ((πR2)/360)α , Where S — area of ​​the circular sector

The circumference and circle got their name in Ancient Greece. Already in ancient times, people were interested in round bodies, so the circle became the crown of perfection. The fact that a round body could move on its own was the impetus for the invention of the wheel. It would seem, what is special about this invention? But imagine if in an instant the wheels disappear from our lives. This invention later gave rise to the mathematical concept of a circle.