Formula for calculating a circular sector. Formulas for the area of ​​a sector of a circle and the length of its arc

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Burning paper is not used in geometry. Geometry and sheet of paper. Pascal. A triangle is cut out of paper. Leaf from a notebook. Among the many possible actions with paper, an important place is occupied by the fact that it can be cut. "History of Geometry" - Ancient Egypt . Middle Ages. "Principles" consists of 13 books. The emergence and development of geometry. In Lyubachevsky geometry there are triangles with pairs. parallel sides Ancient Greece

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“Proof of the Pythagorean Theorem” - The significance of the theorem is that most of the theorems of geometry can be deduced from it or with its help. Algebraic proof. The meaning of the Pythagorean theorem. And now the Pythagorean theorem is true, as in his distant age. The Pythagorean theorem is one of the most important theorems in geometry. Pythagorean theorem. Euclid's proof. “Thales of Miletus” - THALES is an ancient Greek thinker, the founder of ancient philosophy and science. Sometimes it is necessary to measure the distance to an inaccessible object. Determining distance using a match. Thales discovered the length of the year and divided it into 365 days. Thales of Miletus. Thales predicted solar eclipse

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There are 41 presentations in total AND - circle geometric figures , interconnected. there is a boundary broken line (curve),

Definition. A circle is a closed curve, each point of which is equidistant from a point called the center of the circle.

To construct a circle, an arbitrary point O is selected, taken as the center of the circle, and a closed line is drawn using a compass.

If point O of the center of the circle is connected to arbitrary points on the circle, then all the resulting segments will be equal to each other, and such segments are called radii, abbreviated as Latin small or capital letter"er" ( r or R). You can draw as many radii in a circle as there are points in the length of the circle.

A segment connecting two points on a circle and passing through its center is called a diameter. Diameter consists of two radii, lying on the same straight line. Diameter is indicated by the Latin small or capital letter “de” ( d or D).

Rule. Diameter a circle is equal to two its radii.

d = 2r
D=2R

The circumference of a circle is calculated by the formula and depends on the radius (diameter) of the circle. The formula contains the number ¶, which shows how many times the circumference is greater than its diameter. The number ¶ has an infinite number of decimal places. For calculations, ¶ = 3.14 was taken.

The circumference of a circle is denoted by the Latin capital letter “tse” ( C). The circumference of a circle is proportional to its diameter. Formulas for calculating the circumference of a circle based on its radius and diameter:

C = ¶d
C = 2¶r

• Examples
• Given: d = 100 cm.
• Circumference: C=3.14*100cm=314cm
• Given: d = 25 mm.
• Circumference: C = 2 * 3.14 * 25 = 157mm

Circular secant and circular arc

Every secant (straight line) intersects a circle at two points and divides it into two arcs. The size of the arc of a circle depends on the distance between the center and the secant and is measured along a closed curve from the first point of intersection of the secant with the circle to the second.

Arcs circles are divided secant into major and minor, if the secant does not coincide with the diameter, and into two equal arcs, if the secant passes along the diameter of the circle.

If a secant passes through the center of a circle, then its segment located between the points of intersection with the circle is the diameter of the circle, or the largest chord of the circle.

The further the secant is located from the center of the circle, the smaller the degree measure of the smaller arc of the circle and the larger the larger arc of the circle, and the segment of the secant, called chord, decreases as the secant moves away from the center of the circle.

Definition. A circle is a part of a plane lying inside a circle.

The center, radius, and diameter of a circle are simultaneously the center, radius, and diameter of the corresponding circle.

Since a circle is part of a plane, one of its parameters is area.

Rule. Area of ​​a circle ( S) is equal to the product of the square of the radius ( r 2) to the number ¶.

• Examples
• Given: r = 100 cm
• Area of ​​a circle:
• S = 3.14 * 100 cm * 100 cm = 31,400 cm 2 ≈ 3 m 2
• Given: d = 50 mm
• Area of ​​a circle:
• S = ¼ * 3.14 * 50 mm * 50 mm = 1,963 mm 2 ≈ 20 cm 2

If you draw two radii in a circle to different points on the circle, then two parts of the circle are formed, which are called sectors. If you draw a chord in a circle, then the part of the plane between the arc and the chord is called circle segment.

There is no need to learn the area of ​​a sector of a circle and the area of ​​a segment! Dear friends!You've probably looked through a reference book with mathematical formulas more than once, and the thought probably arose: “Is it really possible to learn them all?” I'll tell you what's possible, but why? Why fill your head with a lot of formulas, constantly repeat them, be horrified that you forgot some and repeat them again? No need!

In fact, it is enough to remember a third of all formulas, basic formulas or even less. Next you will understand what we are talking about. All other formulas can be quickly deduced by knowing the basics, applying logic, and remembering the principles to follow.

Let me give you an example: there are 32 reduction formulas; learning them is a pointless exercise. How to quickly remember any of them is outlined in the article “”, take a look.

In this article we will look at how to quickly restore in memory the formulas for the area of ​​a sector of a circle, the area of ​​its segment, and the length of the arc of a circle. It is these formulas that will be needed to solve the series in planimetry, which we will analyze in the next article.So, “basic” formulas, you need to learn and know them!

Area of ​​a circle (formula):

Circumference formula:

Let us depict a sector corresponding to a certain central angle n:

We reason logically: if the area of ​​a circle is S= PR 2 , then the area corresponding to a sector of one degree will be equal to 1/360 of the area of ​​the circle (we know that the entire circle is an angle of 360 degrees), that is

It is further clear that the area of ​​the sector corresponding to the central angle of n degrees is equal to the product of one three hundred and sixtieth of the area of ​​the circle and the central angle n (corresponding to the sector), that is

Here is the formula for the sector area.

Or you can structure your reasoning as follows:

A sector of 1 degree is 1/360 of a circle, respectively, a sector of n degrees is n/360 of a circle. That is, the area of ​​the sector will be equal to the product of the area of ​​the circle and this part:

It's simple. It is necessary to subtract the area of ​​the triangle from the area of ​​the sector (it is designated yellow). The area of ​​a triangle, as we know, is equal to half the product neighboring parties by the sine of the angle between them (you need to know this formula, it is notcomplex). In this case it is:

Means,

So much for the segment area!

Segment area, where central angle more than 180 degrees is found simply:

From the area of ​​the circle, subtract the area of ​​the resulting segment:

The angle 360 ​​– n degrees is the angle that corresponds to the depicted sector (yellow):

That is, in other words, we add the area of ​​the triangle to its area and get the area of ​​the specified segment.

Similarly, we determine the length of the arc of a circle. As already said, the circumference is equal to:

This means that the length of the arc of a circle corresponding to one degree will be equal to one three hundred and sixtieth of 2πR, that is

We get the length of the arc of a circle. Certainly, this information teachers give to students, and you haven’t learned anything so secret. But I’m sure the article will be useful to you.

I repeat that the most important thing is to know the formulas for the area of ​​a circle and the circumference, and then only logic works.

I suggest you look additional lesson Dmitry Tarasov on this topic. Formulas for the length of a circular arc and the area of ​​a sector are considered, where the central angle is given in radian measure.

That's all. I wish you success!!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

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 to 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 . Everything that I am telling you here at length, she does 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 kinds of round and semicircular parts, but more on this in 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 absolutely free program, which performs all of the above calculations, freeing you from having to remember what an arctangent is and where to look for it.

The circle is the main figure in geometry, the properties of which are studied at school in the 8th grade. One of the typical problems involving a circle is to find the area of ​​some part of it, which is called a circular sector. The article provides formulas for the area of ​​a sector and the length of its arc, as well as an example of their use to solve a specific problem.

The concept of circumference and circle

Before giving the formula for the area of ​​a sector of a circle, let’s consider what the indicated figure is. According to mathematical definition, a circle is understood as a figure on a plane, all points of which are equidistant from some single point (center).

When considering a circle, the following terminology is used:

• Radius is a segment drawn from the center point to the curve of the circle. It is usually denoted by the letter R.
• A diameter is a line segment that connects two points on a circle, but also passes through the center of the figure. It is usually denoted by the letter D.
• An arc is a part of a curved circle. It is measured either in units of length or using angles.

The circle is another important figure in geometry; it is a collection of points that is bounded by the curve of a circle.

Area of ​​a circle and circumference

The values ​​noted in the title of the item are calculated using two simple formulas. They are given below:

• Circumference: L = 2*pi*R.
• Area of ​​a circle: S = pi*R 2 .

In these formulas, pi is a certain constant called the Pi number. It is irrational, that is, it cannot be accurately expressed as a simple fraction. The approximate value of Pi is 3.1416.

As can be seen from the above expressions, in order to calculate the area and length it is enough to know only the radius of the circle.

Area of ​​a sector of a circle and length of its arc

Before considering the corresponding formulas, let us recall that angles in geometry are usually expressed in two main ways:

• in sexagesimal degrees, with a complete revolution around its axis being 360 o;
• in radians, which are expressed in fractions of the number pi and are related to degrees by the following equality: 2*pi = 360 o.

A sector of a circle is a figure bounded by three lines: an arc of a circle and two radii located at the ends of this arc. An example of a circular sector is shown in the photo below.

Having gained an idea of ​​what a sector of a circle is, it is easy to understand how to calculate its area and the length of the corresponding arc. From the figure above it can be seen that the arc of the sector corresponds to the angle θ. We know that a complete circle corresponds to 2*pi radians, which means that the formula for the area of ​​a circular sector will take the form: S 1 = S*θ/(2*pi) = pi*R 2 *θ/(2*pi) = θ*R 2 /2. Here the angle θ is expressed in radians. A similar formula for the sector area if the angle θ is measured in degrees will look like: S 1 = pi*θ*R 2 /360.

The length of the arc forming the sector is calculated by the formula: L 1 = θ*2*pi*R/(2*pi) = θ*R. And if θ is known in degrees, then: L 1 = pi*θ*R/180.

Example of problem solution

Using a simple problem as an example, we will show how to use the formulas for the area of ​​a sector of a circle and the length of its arc.

It is known that the wheel has 12 spokes. When the wheel makes one full revolution, it covers a distance of 1.5 meters. What is the area enclosed between two adjacent spokes of the wheel, and what is the length of the arc between them?

As can be seen from the corresponding formulas, in order to use them, you need to know two quantities: the radius of the circle and the angle of the arc. The radius can be calculated based on knowledge of the circumference of the wheel, since the distance it travels in one revolution corresponds exactly to it. We have: 2*R*pi = 1.5, from where: R = 1.5/(2*pi) = 0.2387 meters. The angle between the nearest spokes can be determined by knowing their number. Assuming that all 12 spokes evenly divide the circle into equal sectors, we get 12 identical sectors. Accordingly, the angular measure of the arc between the two spokes is equal to: θ = 2*pi/12 = pi/6 = 0.5236 radians.

We have found all the necessary quantities, now we can substitute them into the formulas and calculate the values ​​​​required by the condition of the problem. We get: S 1 = 0.5236 * (0.2387) 2 /2 = 0.0149 m 2, or 149 cm 2; L 1 = 0.5236*0.2387 = 0.125 m, or 12.5 cm.