What is the volume of the ring

What is a circular ring?

A circular ring is the area between two different circles with the same center.

......A circular ring is generally determined by the inner radius r and the outer radius R.
Then it has an area of ​​A = pi (R²-r²).
The boundary lines (circumference) have a length of 2pi (R + r).

Given the width or thickness d of the ring and the inner radius r, the outer radius R = r + d.
A = pi (2rd + d²) applies to the area.
If the width is significantly smaller than the radius, the simplified formula A = 2pi * rd applies approximately.

Circle of equal area
The circle, which has the same area as the annulus, has a radius of x = sqrt (R²-r²).
The circle we are looking for has a radius of x.
The approach pi * x² = pi (R²-r²) leads to x = sqrt (R²-r²).

The diameter of the circle of the same area is also obtained by drawing a tangent on the inner circle and taking the section within the circular ring as the diameter of the circle.
According to the Pythagorean theorem, R² = x² + r² or x² = R²-r².

The following applies to the circle with radius x:
A = pi * x² = pi * (R²-r²). But that is precisely the area of ​​the circular ring.

Lots of concentric circles
The figure below is known as a target.
......It consists of a series of circular rings of equal thickness. The radii of the concentric circles are r, 2r, 3r, 4r, ...

A applies to the inner circle0= pi * r².
The following applies to the first ring: A.1= pi * [(2r) ²-r²] = 3 * pi * r².
The following applies to the second ring: A.2= pi * [(3r) 2- (2r) 2] = 5 * pi * r².

For the nth ring, A appliesn= pi * [((n + 1) r) 2- (nr) 2] = pi * (2n + 1) r².

Circular rings from within and around
If you draw the circumferences and incircles of the regular polygons, you get circular rings.
The first polygons and some data follow.

It mean
a: side length of a polygon, R: radius of the circumference, r: radius of the inscribed circle, A: area of ​​the circular ring.

It is astonishing that the area of ​​all circular rings is only determined by the side length a of the polygon and is independent of the radii R and r.

Hollow cylinderTop
The hollow cylinder is a body that is formed by two concentric cylinders.

The volume of the hollow cylinder is equal to V = pi * (R²-r²) * h. The surface is made up of the jackets of the inner and outer cylinders and the circular rings above and below:
O = 2 * pi * R * h + 2 * pi * r * h + 2 * pi * (R²-r²) = 2 * pi * (Rh + rh + R²-r²).
For a ring, h << r.
If d = R-r << h, a pipe or a hose is created. - If h << R and r << R, a disk is created.

Torus   Top
The torus is also called a ring body, circular bead, lifebuoy and, in special designs, a wreath.

......A torus is created when a vertical circle rotates around a vertical axis outside the circle. The circle and axis lie in one plane.
Here r is the radius of the circle and R is the distance between the center of the circle and the axis of rotation.

Above left the torus is shown in section, below in the top view.

One can imagine that the torus consists of any number of circular disks of any thickness, which then form the ring. If you bend it to a cylinder, this has the base area pi * r² and the height 2 * pi * R.
For the volume of the torus we get V = (pi * r²) * (2 * pi * R) = 2pi²r²R.
For the surface, O = (2 * pi * r) * (2 * pi * R) = 4pi²rR. That is the jacket of the imaginary cylinder.
The two Guldin's rules provide the exact mathematical derivation of these two formulas.

The adjoining torus shows the parametric representation

x = cos (s) [3 + cos (t)]
y = sin (s) [3 + cos (t)]
z = sin (t)

with 0 <= s <2 * Pi and 0 <= t <2 * Pi.

In the meantime (September 2010) the torus has its own website on my homepage. You can find more there.

Intertwined rings Top
Double ring

......Two rings can be connected in such a way that they cannot be separated again without breaking one ring. This arrangement of two rings is considered a symbol of fidelity and marriage.
Several rings connected in this way form a chain. In general, it is self-contained, such as the necklace.

......You can connect three rings to form a triangle. At first glance, the two figures are the same. Imagine removing a ring, the rings remain connected on the left, the rings on the right are then individual.
The arrangement on the right is called the Borromeo Link.

Krupp rings
......With a figure made up of three rings, one thinks of the croup rings. They refer to Alfred Krupp's invention of the "seamlessly forged and rolled railway wheel tire".
The rings are not interwoven. The upper ring is in front of the two lower rings.
Today the rings are no longer separated. (More at Thyssen Krupp, URL below)

Audi rings
The Audi logo consists of a straight chain of four rings.

The four rings stand for the brands Audi, DKW, Horch and Wanderer, which were combined to form AUTO UNION in 1932.

The Olympic Rings
......Everyone knows the official emblem of the IOC, the five entwined Olympic rings.
They are created by laying the straight chain (1) in a W-shape (2).
The colors of the rings (and the background color white) are chosen so that each country can find one of its colors in the Olympic colors.
They do not officially stand for the five continents. Nevertheless, an assignment has become established:
blue europe
yellow asia
black africa
green australia
red america
It's no coincidence that I didn't dye the rings. (See article by lawyer Thomas Engels, URL below)

Wedding ring Top

......From a mathematical point of view, the wedding ring, like the torus, is a solid of revolution. The cross-section is not a circle, but an elongated figure with a straight line inside.
Due to the straight inner border, the ring fits better and there is space for an engraving.
It is a beautiful custom for a married couple to express their close bond by wearing two identical rings on their ring finger.
If the husband dies, the widow wears both wedding rings side by side.

We Germans wear the wedding ring on right Ring finger. This is how you can recognize us abroad (also).
In the husband's ring, the wife's first name is usually written, conversely his name is in her ring.
In addition, the day of the marriage is shown in both rings. This is a memory aid for husbands ;-).

There are numerous games or gimmicks with rings. Here is a small selection.

......The wedding ring is an excellent top.

If you are right-handed, you grasp the ring with the index finger of your right hand and the thumb of your left hand and give it two torques with a jerk. When you let go, both forces must be about the same so that it stays in one place and rotates around the vertical axis of symmetry.

The ring spins for a long time and so fast that you can see a ball.

......If you put a wedding ring on a piece of paper and let light fall on it at an angle from above, for example sunlight, it will be reflected on the inner wall of the ring. The reflected rays illuminate the interior somewhat and surprisingly, the boundary line towards the center has the shape of a three. That is the catacaustics.
The best way to explain the creation of the catacustic acoustics is through a series of drawings.
The situation is simplified to a flat problem: the cylindrical inner surface of the ring becomes a circle.
Image 1: Parallel rays of light fall from the left into the ring, which then acts as a spherical mirror. How are they reflected?
Picture 2: A ray of light is traced to represent the parallel rays. It is reflected at a point where a "mirror" has the direction of the tangent plane. This is the tangent to the circle in the drawing. It can be found by drawing the perpendicular to the contact radius r. According to the first part of the law of reflection, the perpendicular (radius), the incident and the reflected ray lie in one plane. This is the plane of the drawing here. According to the 2nd part of the law of reflection, the angle of reflection is as large as the angle of incidence. This will find the reflected beam.
Picture 3: According to this rule, all reflected rays are drawn. The result is image 3, which reproduces the photograph above well.
Picture 4: Even more can be read from the drawing: The parallel rays near the axis meet at a (focal) point F. The focal length f = FS is equal to half the circle radius.

......With a ring hanging on a thin thread, you can tell in a woman whether the next child will be a boy or a girl.

The pendulum is hung over the woman's wrist. The ring itself must hover over the pulse (picture). If you keep your hand still above, the pendulum will still swing after a certain time. If it swings back and forth in one plane, a boy announces himself. When the ring circles, a girl comes.

If you want, you can believe in it ;-).

An explanation is interesting. I believe that the upper hand cannot be held still, if only because the heart is beating rhythmically. The pendulum receives unconscious impulses. So it's likely a resonance phenomenon.

......By the way: It reminds of old children's fun. In spring, the leaf of a broad-planted plantain is torn out in such a way that the stem (new spelling!) Is separated in the middle. Then the petiole has threads at the bottom that indicate the future number of children and, through different lengths, the distance between the children. In this case three children will come ;-).

Ring rates
Ring guessing is an old parlor game (see also Book 1).
......In a society of up to nine people, a person sticks himself unnoticed by an outsider, let's call him A, a ring on some phalanx. Everyone is hiding their fingers. Person A has to find out who is wearing the ring. It has to name three numbers, because the location of a ring is determined beforehand by three numbers: These are the number of the person (1 to 9), the number of a person's finger (1 to 4) and the number of the phalanx of a finger (1 to 3).

A presents society with a math problem: double the person's number, add 5, multiply by 5, add the number of the finger, multiply by 10, add the number of the phalanx and finally subtract 250.

To the amazement of society, A mentions the place. That is the result of the calculation. This is a three-digit number consisting of the number of the person, the number of the finger and the number of the phalanx.

General: The number of the person with the ring is a, the number of the finger is b and the number of the phalanx is c. The calculation is then [(2 * a + 5) * 5 + b] * 10 + c-250 = 100 * a + 10 * b + c.

Rings of numbers Top
If you determine the decimal number of the fraction 1/7 (for example by dividing 1:7), you get the purely periodic representation 0.142857142857142857142857142857142857142857 ... This sequence of digits can be continued as far as you want. The numbers 142857 repeat themselves and form the "period".

......It is clear when the numbers for the period 142857 are arranged in a circle.
A ring is then created that is run through again and again in a clockwise direction if the accuracy of the decimal number is increased.
Furthermore, 2/7 = 0.285714 ..., 3/7 = 0.428571 ..., 4/7 = 0.571428 ..., 5/7 = 0.714285 ..., 6/7 = 857142.
Here, too, the period has the length 6 and the same digits appear. You can even use the same ring as with 1/7, only that the entry points into the circle are different.
Instead of 1/7 you can also choose 1/17. The period, understood as a number, is called the phoenix number. It is undecided whether there are an infinite number of phoenix numbers (2, page 188).

(1) Torus (2) Hollow cylinder (3) Circular ring

Rings, rings, rings Top

Amulet ring
Apple picker
Bangle, bangle
Benzene ring
Bishop's Ring
Boxing ring ("ring with 4 corners")
Bull ring
"The ring closes"
Sealing ring
Mason ring
Bicycle or car inner tube
Fresnel ring lens
Neck ring
Lord of the rings
Witch ring
Chicken ring
Hoola hoop
Iffland Ring
Iris of the eye
Annual ring
Core memory
Circular waves
Magic rings
Tit ring
Moon ring
Coin € 1 or € 2
Nose ring
Smoke ring
RCDS (Ring of Christian Democratic Students)
Ring accelerator
Ring binder
Ring of the Nibelungs
Ring of saturn
Ring through which a lion jumps
Rings as gymnastics equipment
Ringelpiez you can touch
Ring dance
Curly tail
Striped socks
Circles under the eyes
Ring manhunt
Ring finger
Ring Mountains
Ring trench
Ring line
Ring nebula
Curtain wall
Sphincter muscle
Ring furnace
Ring parabola
Ring rates
Ring broadcast
Ring road
Ring current effect
Ring tennis
Lecture series
Ring wall
Key ring
Jewelry ring
Swim ring
Signet ring
Token ring
Wedding ring
Bird ringing
White ring
Magic ring.

Rings on the internet Top


Herwig Hauser (Faculty of Mathematics, University of Vienna)
Torus animation

Homepage Brothers Grimm
Catacaustics, lecture by Dr. C. Ucke

Micha Peteler
Forge ring

Lawyer Thomas Engels
The proverbial game with fire when using the Olympic rings

Thyssen Krupp

Circular ring, torus, toroidal core, cylinder (geometry), Olympic rings, Borromean rings, donut, phoenix number


Eric W. Weisstein (MathWorld)
Torus, Double Torus, TripleTorus, Impossible Torus

John Rausch (Puzzle World)
Puzzle Ring Solution

Richard Parris (freeware program WINPLOT)
The official website is closed. Download the German program e.g. from heise

Annulus (mathematics), Torus, Toroid (geometry), Cylinder (geometry), Olympic symbols, Borromean rings, Donut, Cyclic number

credentials Top
(1) Hans Nicklisch (Ed.): Schlag nach, Natur, Leipzig 1952 (page 41/42)
(2) Endre Hódi (Ed.): Mathematisches Mosaik, Cologne

I would like to thank Jürgen Dornieden for a clarification on the Olympic rings.

Feedback: Email address on my main page

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© 2002, edited 2018, Jürgen Köller