Initially it looks like this:
Figure 463.1. a) the existing arc, b) determination of the 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 of such a problem was considered in sufficient detail in the article "Calculation of an arched lintel", therefore, here I will only give the basic formulas:
tg( a/4) = 2H/L (278.1.2)
A/4 = arctan( 2H/L)
R = H/(1 - cos( a/2)) (278.1.3)
As you can see, from the point of view of mathematics, there are no problems with determining the radius of a circle. This method allows you to determine the value of the radius of the arc with any possible accuracy. This is the main merit this method.
Now let's talk about the disadvantages.
The problem with this method is not even that you need to remember formulas from school course geometries, successfully forgotten many years ago - in order to recall the formulas - there is the Internet. And here is a calculator with the function arctg, arcsin, and so on. Not every user has one. And although the Internet also successfully solves this problem, we should not forget that we are solving a rather applied problem. Those. it is far from always necessary to determine the radius of a circle with an accuracy of 0.0001 mm, an accuracy of 1 mm can be quite acceptable.
In addition, in order to find the center of the circle, you need to extend the height of the segment and set aside a distance equal to the radius on this straight line. Since in practice we are dealing with non-ideal measuring instruments, a possible error in marking should be added to this, it turns out that the lower the height of the segment in relation to the length of the chord, the greater the error in determining the center of the arc.
Again, we should not forget that we are not considering an ideal case, i.e. This is how we immediately called the curve an arc. In fact, it can 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 myself often use, because this method is much faster and easier to determine the radius of a circle, although the accuracy is much less.
The second method for determining the radius of the arc (method of successive approximations)
So let's continue with the current situation.
Since we still need to find the center of the circle, to begin with, from the points corresponding to the beginning and end of the arc, we draw at least two arcs of arbitrary radius. A straight line will pass through the intersection of these arcs, 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 from the indicated points not along one arc, 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 considered arc 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 considered arc is lower on the straight line drawn through the intersection of the arcs and the middle of the chord. If less, then the desired center of the arc is higher on the straight line.
Based on this, the next point is taken on the straight line, presumably corresponding to the center of the arc, and the same measurements are made from it. Then the next point is taken and the measurements are repeated. With each new point, the difference in measurements will be less and less.
That's actually all. Despite such a lengthy and intricate description, it takes 1-2 minutes to determine the radius of the arc in this way with an accuracy of 1 mm.
Theoretically, it looks something like this:
Figure 463.2. Determining the center of the arc by the method of successive approximations.
But in practice, something like this:
Photo 463.1. Marking a workpiece of complex shape with different radii.
I’ll just add here that sometimes you have to find and draw several radii, because there are so many things mixed up in the photo.
The circle, its parts, their sizes and ratios are things that the jeweler constantly encounters. Rings, bracelets, castes, tubes, balls, spirals - a lot of round things have to be done. How can you calculate all this, especially if you were lucky enough to skip geometry lessons at school? ..
Let's first look at what the circle has parts and what they are called.
- A circle is a line that encloses a circle.
- An arc is part of a circle.
- A radius is a line segment that connects the center of a circle to a point on the circle.
- A chord is a line segment that connects 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 of interest to us and their designations:
Now let's see what tasks related to the parts of the 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 the plane, you need to find out its size in projection after bending into an arc. Given the length of the arc and the diameter, find the length of the chord.
- Find out the height of a part obtained by bending a flat workpiece into an arc. Initial data options: arc length and diameter, arc length and chord; find the height of the segment.
Life will prompt other examples, and I gave these only to show the need to set any two parameters to find all the others. That's what we're going to do. Namely, we take five segment parameters: 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 burden the reader in vain, I will not give detailed solutions, but will only give results in the form of formulas (I will discuss those cases where there is no formal solution along the way).
And one more remark: about units of measurement. All quantities, except for 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 \u200b\u200bwill 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 practice shows, people who design something round are not inclined to measure angles in radians. The phrase "the angle of pi by four" confuses many, while the "angle of forty-five degrees" is understandable to everyone, since it is only five degrees above the norm. However, in all formulas there will be one more angle - α - as an intermediate value. In terms of meaning, this is half the central angle, measured in radians, but you can safely not delve into this meaning.
1. Diameter D and arc length L are given
; chord length 
;
segment height 
; central corner 
.
2. Diameter D and chord length X are given
; arc length;
segment height ; central corner 
.
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 with the angle in the above formulas.
3. Diameter D and central angle φ are given
; arc length;
chord length ; segment height .
4. Given the diameter D and the height of the segment H
; arc length;
chord length ; central corner .
6. Given the length of the arc L and the central angle φ
; diameter ;
chord length ; segment height .
8. Given the length of the chord 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 corner .
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 the length of the arc L and the length of the chord 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 to take a mandrel (crossbar)?
This task is reduced to solving the equations: 
; - in option 5
; - in option 7
and although they are not solved analytically, they are easily solved programmatically. And I even know where to get such a program: on this very site, under the name . Everything that I tell here at length, she does in microseconds.
To complete the picture, let's add to the results of our calculations the circumference and three values of areas - a circle, a sector and a segment. (The areas will help us a lot when calculating the mass of any round and semicircular parts, but more on that 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 of the existence of absolutely free program, which performs all the above calculations, freeing you from having to remember what the arc tangent is and where to look for it.
Definition of 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.
ChordThis is a segment that lies inside the circle and connects two arbitrarily chosen points on it.
When cutting off part of the circle with a chord, two figures can be considered: 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 the sector of the circle and this isosceles triangle.
The area of a segment can be found in several ways. Let's dwell on them in more detail.
The formula for the area of a segment of a circle in terms of the radius and length of the arc 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- circle radius;
s s s- arc length;
h h h- height of an isosceles triangle;
a a a is the length of the base of this triangle.
A circle is given, its radius, numerically equal to 5 (see), the height that is drawn to the base of the triangle, equal to 2 (see), the length of the arc is 10 (see). 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 lack only the base of the triangle. Let's find it by 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 square
The 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- circle radius;
α\alpha α
is the central angle between two radii subtending the chord, measured in radians.
Find the area of a segment of a circle 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 the π\pi π
radian equals 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:
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 square
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