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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 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 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 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 mathematical value of area has been known since the time ancient Greece. Even in those distant times, the Greeks found out that an area is a continuous part of a surface, which is limited on all sides by a closed contour. This is a numerical value that is measured in square units. Area is a numerical characteristic of both flat geometric figures (planimetric) and the surfaces of bodies in space (volumetric).

Currently, it is found not only in the school curriculum in geometry and mathematics lessons, but also in astronomy, everyday life, construction, design development, manufacturing and many other human subjects. Very often we resort to calculating the areas of segments using personal plot when designing a landscape area or during renovation work of ultra-modern room design. Therefore, knowledge of methods for calculating various areas will be useful always and everywhere.

To calculate the area of ​​a circular segment and a sphere segment, you need to understand the geometric terms that will be needed during the computational process.

First of all, a segment of a circle is a fragment of a flat figure of a circle, which is located between the arc of a circle and the chord cutting it off. This concept should not be confused with the sector figure. These are completely different things.

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

The central angle is formed between two segments - radii. It is measured in degrees by the arc on which it rests.

A segment of a sphere is formed when a part is cut off by some plane. In this case, the base of the spherical segment is a circle, and the height is the perpendicular emanating from the center of the circle to the intersection with the surface of the sphere. This intersection point is called the vertex of the ball segment.

In order to determine the area of ​​a sphere segment, you need to know the cut-off circle and the height of the spherical segment. The product of these two components will be the area of ​​the sphere segment: S=2πRh, where h is the height of the segment, 2πR is the circumference, and R is the radius of the great circle.

In order to calculate the area of ​​a circle segment, you can resort to the following formulas:

1. To find the area of ​​a segment by the most in a simple way, it is necessary to calculate the difference between the area of ​​the sector in which the segment is inscribed, and whose base is the chord of the segment: S1=S2-S3, where S1 is the area of ​​the segment, S2 is the area of ​​the sector and S3 is the area of ​​the triangle.

You can use an approximate formula for calculating the area of ​​a circular segment: S=2/3*(a*h), where a is the base of the triangle or h is the height of the segment, which is the result of the difference between the radius of the circle and

2. The area of ​​a segment different from a semicircle is calculated as follows: S = (π R2:360)*α ± S3, where π R2 is the area of ​​the circle, α is the degree measure of the central angle, which contains the arc of the circle segment, S3 is the area of ​​the triangle that was formed between the two radii of the circle and the chord, which has an angle at the central point of the circle and two vertices at the points of contact of the radii with circle.

If angle α< 180 градусов, используется знак минус, если α >180 degrees, plus sign applied.

3. You can calculate the area of ​​a segment using other methods using trigonometry. As a rule, a triangle is taken as a basis. If the central angle is measured in degrees, then the following formula is acceptable: S= R2 * (π*(α/180) - sin α)/2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle.

4. To calculate the area of ​​a segment using trigonometric functions, you can use another formula, provided that the central angle is measured in radians: S= R2 * (α - sin α)/2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle.

Defining a Circle Segment

Segment- This geometric figure, which is obtained by cutting off part of the circle with a chord.

Online calculator

This figure is located between the chord and the arc of the circle.

Chord

This is a segment lying inside a circle and connecting two arbitrarily chosen points on it.

When cutting off part of a circle with a chord, you can consider two figures: this is our segment and an isosceles triangle, the sides of which are the radii of the circle.

The area of ​​a segment can be found as the difference between the areas of a sector of a circle and this isosceles triangle.

The area of ​​a segment can be found in several ways. Let's look at them in more detail.

Formula for the area of ​​a circle segment using the radius and arc length of the circle, the height and base of the triangle

S = 1 2 ⋅ R ⋅ s − 1 2 ⋅ h ⋅ a S=\frac(1)(2)\cdot R\cdot s-\frac(1)(2)\cdot h\cdot aS=2 1 ​ ⋅ R⋅s −2 1 ​ ⋅ h⋅a

R R R- radius of the circle;
s s s- arc length;
h h h- height of an isosceles triangle;
a a a- the length of the base of this triangle.

Example

Given a circle, its radius is numerically equal to 5 (cm), the height, which is drawn to the base of the triangle, is equal to 2 (cm), the length of the arc is 10 (cm). Find the area of ​​a circle segment.

Solution

R=5 R=5 R=5
h = 2 h=2 h =2
s = 10 s=10 s =1 0

To calculate the area, we only need the base of the triangle. Let's find it using the formula:

A = 2 ⋅ h ⋅ (2 ⋅ R − h) = 2 ⋅ 2 ⋅ (2 ⋅ 5 − 2) = 8 a=2\cdot\sqrt(h\cdot(2\cdot R-h))=2\cdot\ sqrt(2\cdot(2\cdot 5-2))=8a =2 ⋅ h ⋅ (2 ⋅ R − h )​ = 2 ⋅ 2 ⋅ (2 ⋅ 5 − 2 ) ​ = 8

Now you can calculate the area of ​​the segment:

S = 1 2 ⋅ R ⋅ s − 1 2 ⋅ h ⋅ a = 1 2 ⋅ 5 ⋅ 10 − 1 2 ⋅ 2 ⋅ 8 = 17 S=\frac(1)(2)\cdot R\cdot s-\frac (1)(2)\cdot h\cdot a=\frac(1)(2)\cdot 5\cdot 10-\frac(1)(2)\cdot 2\cdot 8=17S=2 1 ​ ⋅ R⋅s −2 1 ​ ⋅ h⋅a =2 1 ​ ⋅ 5 ⋅ 1 0 − 2 1 ​ ⋅ 2 ⋅ 8 = 1 7 (see sq.)

Answer: 17 cm sq.

Formula for the area of ​​a circle segment given the radius of the circle and the central angle

S = R 2 2 ⋅ (α − sin ⁡ (α)) S=\frac(R^2)(2)\cdot(\alpha-\sin(\alpha))S=2 R 2 (α − sin(α))

R R R- radius of the circle;
α\alpha α - the central angle between two radii subtending the chord, measured in radians.

Example

Find the area of ​​a circle segment if the radius of the circle is 7 (cm) and the central angle is 30 degrees.

Solution

R=7 R=7 R=7
α = 3 0 ∘ \alpha=30^(\circ)α = 3 0

Let's first convert the angle in degrees to radians. Because π\pi π A radian is equal to 180 degrees, then:
3 0 ∘ = 3 0 ∘ ⋅ π 18 0 ∘ = π 6 30^(\circ)=30^(\circ)\cdot\frac(\pi)(180^(\circ))=\frac(\pi )(6)3 0 = 3 0 1 8 0 π = 6 π radian. Then the area of ​​the segment is:

S = R 2 2 ⋅ (α − sin ⁡ (α)) = 49 2 ⋅ (π 6 − sin ⁡ (π 6)) ≈ 0.57 S=\frac(R^2)(2)\cdot(\alpha- \sin(\alpha))=\frac(49)(2)\cdot\Big(\frac(\pi)(6)-\sin\Big(\frac(\pi)(6)\Big)\Big )\approx0.57S=2 R 2 (α − sin(α)) =2 4 9 ​ ⋅ ( 6 π ​ − sin ( 6 π ) ) 0 . 5 7 (see sq.)

Answer: 0.57 cm sq.

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The area of ​​a circular segment is equal to the difference between the area of ​​the corresponding circular sector and the area of ​​the triangle formed by the radii of the sector corresponding to the segment and the chord limiting the segment.

Example 1

The length of the chord subtending the circle is equal to the value a. The degree measure of the arc corresponding to the chord is 60°. Find the area of ​​the circular segment.

Solution

A triangle formed by two radii and a chord is isosceles, so the altitude drawn from the vertex of the central angle to the side of the triangle formed by the chord will also be the bisector of the central angle, dividing it in half, and the median, dividing the chord in half. Knowing that the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse, we can calculate the radius:

Sin 30°= a/2:R = 1/2;

Sc = πR²/360°*60° = πa²/6

S▲=1/2*ah, where h is the height drawn from the vertex of the central angle to the chord. According to the Pythagorean theorem h=√(R²-a²/4)= √3*a/2.

Accordingly, S▲=√3/4*a².

The area of ​​the segment, calculated as Sreg = Sc - S▲, is equal to:

Sreg = πa²/6 - √3/4*a²

By substituting a numerical value for the value of a, you can easily calculate the numerical value of the segment area.

Example 2

The radius of the circle is equal to a. The degree measure of the arc corresponding to the segment is 60°. Find the area of ​​the circular segment.

Solution:

The area of ​​the sector corresponding to a given angle can be calculated using the following formula:

Sc = πa²/360°*60° = πa²/6,

The area of ​​the triangle corresponding to the sector is calculated as follows:

S▲=1/2*ah, where h is the height drawn from the vertex of the central angle to the chord. By the Pythagorean theorem h=√(a²-a²/4)= √3*a/2.

Accordingly, S▲=√3/4*a².

And finally, the area of ​​the segment, calculated as Sreg = Sc - S▲, is equal to:

Sreg = πa²/6 - √3/4*a².

The solutions in both cases are almost identical. Thus, we can conclude that to calculate the area of ​​a segment in the simplest case, it is enough to know the value of the angle corresponding to the arc of the segment and one of two parameters - either the radius of the circle or the length of the chord subtending the arc of the circle forming the segment.



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