Calculator truncated cone with offset bases. How to make a scan - pattern for a cone or truncated cone of the specified sizes

Sometimes the task occurs - make a protective umbrella for an exhaust or chimney, an exhaust deflector for ventilation, etc. But before proceeding with the manufacture, you need to make a pattern (or scan) for the material. On the Internet there are all sorts of programs to calculate such sweeps. However, the task is so easy to solve that you will quickly calculate it using the calculator (in the computer) than you will search, download and deal with these programs.

Let's start with a simple version - a simple cone scan. The easiest way to explain the principle of calculating the pattern on the example.

Suppose we need to make a cone with a diameter D cm and the height of H centimeters. It is absolutely clear that a circle with a cut segment will act as a workpiece. Two parameters are known - diameter and height. According to the Pythagore Theorem, we calculate the diameter of the circle of the workpiece (do not confuse with the radius ready cone). Half of diameter (radius) and the height form a rectangular triangle. Therefore:

So, now we know the radius of the workpiece and can cut the circle.

Calculate the angle of the sector to be cut from the circle. We argue as follows: the diameter of the workpiece is 2r, it means that the circumference is equal to PI * 2 * R - i.e. 6.28 * R. Denote her L. The circumference is complete, i.e. 360 degrees. And the length of the circle of the finished cone is equal to p * d. Denote her lm. It is natural, less than the length of the circumference of the workpiece. We need to cut a segment with an arc length of equal difference of these lengths. Apply the rule of relation. If 360 degrees give us a complete circumference of the workpiece, then the desired angle should give the length of the circle of the finished cone.

From the ratio formula, we obtain the angle of the angle X. And the cut sector is found by subtracting 360 - H.

From a round blank with a radius R, you need to cut the sector with an angle (360s). Do not forget to leave a small strip of material for the allen (if the cone fastening is a mustache). After connecting the parties to the cut sector, we obtain a cone of the specified size.

For example: we need a cone for an exhaust pipe umbrella with a height of (H) 100 mm and a diameter (D) 250 mm. According to the Pythagore formula, we obtain the blocking radius - 160 mm. And the length of the circumference of the workpiece, respectively, 160 x 6.28 \u003d 1005 mm. At the same time, the length of the circumference of the desired cone is 250 x 3,14 \u003d 785 mm.

Then we get that the ratio of the angles will be: 785/1005 x 360 \u003d 281 degrees. Accordingly, the sector 360 - 281 \u003d 79 degrees should be cut.

Calculation of the workpiece pattern for a truncated cone.

Such a detail is needed in the manufacture of adapters from one diameter to another or for Volpert-Grigorovich or Hangzhenkov. They are used to improve thrust in a chimney or ventilation pipe.

The task is slightly complicated by the fact that we are unknown the height of the entire cone, but only its truncated part. In general, the original numbers here are three: the height of the truncated cone H, the diameter of the lower opening (base) d, and the diameter of the upper hole Dm (in the scene of the total cone). But we resort to the same simple mathematical constructions based on the theorem of Pythagora and similarity.

In fact, it is obvious that the value (D-Dm) / 2 (half of the difference in diameters) will relate to the height of the truncated cone n as well as the radius of the base to the height of the entire cone, as if it were not truncated. We find a complete height (P) from this ratio.

(D - DM) / 2H \u003d D / 2P

Hence P \u003d D x H / (D-DM).

Now knowing the overall height of the cone, we can reduce the solution to the previous one. Calculate the blank scan as it were for a complete cone, and then "subtract" from it the scan of its upper, unnecessary parts to us. And we can calculate the proportions of the workpiece.

We get on the Pythagore theorem a greater radius of the workpiece - RZ. This is a square root from the sum of the p and d / 2 squares.

A smaller RM radius is a square root of squares (P-H) and DM / 2.

The length of the circumference of our billet is 2 x Pi x Rz, or 6.28 x Rz. And the length of the circumference of the base of the cone - PI X D, or 3.14 x D. The ratio of their lengths and give the ratio of the corners of the sectors, if we accept that the full angle in the workpiece is 360 degrees.

Those. X / 360 \u003d 3,14 x D / 6.28 x Rz

Hence X \u003d 180 x D / Rz (this is an angle that needs to be left to get the length of the base circumference). And it is necessary to cut 360 - x.

For example: we need to make a truncated cone with a height of 250 mm, the diameter base is 300 mm, the diameter of the upper opening 200 mm.

We find the height of the total cone p: 300 x 250 / (300 - 200) \u003d 600 mm

By t. Pythagora find the external radius of the workpiece RZ: the root square from (300/2) ^ 2 + 6002 \u003d 618.5 mm

By the same theorem we find a smaller RM radius: square root from (600 - 250) ^ 2 + (200/2) ^ 2 \u003d 364 mm.

Determine the angle of the sector of our workpiece: 180 x 300 / 618.5 \u003d 87.3 degrees.

On the material, black arcs with a radius of 618.5 mm, then from the same center - arc with a radius of 364 mm. The angle of arc may have about 90-100 degrees of disclosure. We carry out radii with an angle of disclosure 87.3 degrees. Our workpiece is ready. Do not forget to allow the docking of the edges if they are connected to the brass.

The surface of the cone surface is a flat figure obtained by combining the side surface and the base of the cone with some plane.

Scanning Options:

Scan of direct circular cone

The scan of the side surface of the direct circular cone is a circular sector, the radius of which is equal to the length of the forming conical surface L, and the central angle φ is determined by the formula φ \u003d 360 * R / L, where R is the radius of the circumference of the base of the cone.

In a number of objectives of the descriptive geometry, the preferred solution is an approximation (replacement) of the cone inscribed in it the pyramid and the construction of an approximate sweep, which is convenient to apply lines lying on the conical surface.

Algorithm of construction

1. Enter the polygonal pyramid into the conical surface. The greater the side faces of the inscribed pyramid, the more accurate correspondence between the actual and approximate scan.
2. We build the scan of the side surface of the pyramid in the way of triangles. Points belonging to the base of the cone connect the smooth curve.

Example

In the figure below in a direct circular cone, the correct hexagonal sabcdef pyramid is inscribed, and the approximate scan of its side surface consists of six isceived triangles - the faces of the pyramid.

Consider the triangle S 0 a 0 B 0. The length of its sides s 0 a 0 and s 0 b 0 is equal to the resulting L conical surface. The value of A 0 B 0 corresponds to the length A'B '. To build a triangle S 0 a 0 B 0 in an arbitrary place of drawing, we lay the segment S 0 a 0 \u003d L, after which it from the points S 0 and a 0 we carry out the circumference with the radius S 0 B 0 \u003d L and A 0 B 0 \u003d A'B ' respectively. Connect the point of crossing the circles B 0 with points A 0 and S 0.

The faces S 0 B 0 C 0, S 0 C 0 D 0, S 0 D 0 E 0, S 0 E 0 F 0, S 0 F 0 A 0 SABCDEF pyramids are similar to a triangle S 0 a 0 B 0.

Points a, b, c, d, e and f lying at the base of the cone connecting the smooth curve - the arc of the circle, the radius of which is equal to L.

Scan of oblique cone

Consider the order of constructing the scan of the side surface of the inclined cone by the method of approximation (approximation).

Algorithm

1. We enter the hexagon of 123456 in the base circumference. We connect points 1, 2, 3, 4, 5, and 6 to the vertex S. Pyramid S123456, constructed in this way, with some degree of approximation is the replacement of the conical surface and is used in this capacity in further buildings.
2. We determine the natural values \u200b\u200bof the ribs of the pyramids using a method of rotation around the projection direct: In the example, an axis I is used, perpendicular to the horizontal plane of projections and passing through the vertex S.
   So, as a result of rotation of the Rib S5, its new horizontal projection S'5 '1 occupies the position at which it is parallel to the frontal plane π 2. Accordingly, S'''5 '' 1 is a genuine value of S5.
3. We build the scan of the side surface of the pyramid S123456, consisting of six triangles: S 0 1 0 6 0, S 0 6 0 5 0, S 0 5 0 4 0, S 0 4 0 3, S 0 3 0 2, S 0 2 0 1 0. The construction of each triangle is performed in three sides. For example, △ s 0 1 0 6 0 Length S 0 1 0 \u003d S''1 '' 0, S 0 6 0 \u003d S''6 '' 1, 1 0 6 0 \u003d 1'6 '.

The degree of conformity of the approximate sweep valid depends on the number of edges of the inscribed pyramid. The number of faces are chosen based on the convenience of reading the drawing, the requirements for its accuracy, the presence of characteristic points and lines to be transferred to the scan.

Transfer line from the surface of the cone to the scan

Line N, lying on the surface of the cone, is formed as a result of its intersection with some plane (figure below). Consider the algorithm for the construction of the N line on the scan.

Algorithm

1. We find the projection of the points A, B and C, in which the line N crosses the ribs inscribed in the cone pyramid S123456.
2. We determine the natural value of segments of SA, SB, SC by the method of rotation around the projection direct. In the example of SA \u003d S'''a '', SB \u003d S'''B '' 1, Sc \u003d S'''c '' 1.
3. We find the position of points a 0, b 0, C 0 on the relevant ribs of the pyramids, laying on the scan of the segment S 0 a 0 \u003d S'''''' '', S 0 B 0 \u003d S''B '' 1, s 0 C 0 \u003d S''c '' 1.
4. Connect points a 0, b 0, C 0 smooth line.

Scan of truncated cone

The method of constructing a straight circular truncated cone described below is based on the principle of similarity.

Geometry as science has formed in ancient Egypt and has reached a high level of development. The famous philosopher Plato founded the Academy, where close attention was paid to the systematization of existing knowledge. Cone as one of the geometric figures is first mentioned in the well-known treatise Euclida "Beginning". Euclid was familiar with the works of Plato. Now there are few people know that the word "cone" in the Greek denotes "pine cones". Greek mathematician Euclid, who lived in Alexandria, is rightfully considered the founder of geometric algebra. The ancient Greeks not only became successors of the knowledge of the Egyptians, but also significantly expanded the theory.

The history of the definition of cone

Geometry as science appeared from the practical requirements of construction and observations of nature. Gradually, experienced knowledge was generalized, and the properties of some bodies were proved through others. The ancient Greeks introduced the concept of axioms and evidence. The axiom is called the approval obtained practically and not requiring evidence.

In his book, Euclid led the definition of a cone as a figure, which is obtained by the rotation of the rectangular triangle around one of the cathets. It also owns the main theorem determining the volume of the cone. And I proved this theorem ancient Greek mathematician Evdox Book.

Another mathematician of ancient Greece, Apollonium Perga, who was a disciple Euclida, developed and outlined the theory of conical surfaces in his books. It owns the definition of a conical surface and the sequential to it. Schoolchildren of our days are studying Euclidean geometry, which remained the main theorems and definitions from ancient times.

Main definitions

The direct circular cone is formed by the rotation of the rectangular triangle around one category. As can be seen, the concept of cone has not changed since the Euclideus.

The AS rectangular triangle AOS hypotenuse during rotation around the OS category forms the side surface of the cone, therefore it is called forming. A triangle OS rolls simultaneously into the height of the cone and its axis. Point S becomes a vertex cone. Carpet AO, describing the circle (base), turned into a radius of the cone.

If there is a plane over the top and the cone axis on top, then you can see that the resulting axial cross section is a chain triangle, in which the axis is the height of the triangle.

where C. - base circumference l. - length of the forming cone, R. - Radius of the base.

Cone volume calculation formula

To calculate the volume of the cone uses the following formula:

where S is the area of \u200b\u200bthe base of the cone. Since the base is a circle, its area is calculated like this:

This implies:

where V is the volume of the cone;

n is a number equal to 3.14;

R is the radius of the base corresponding to the segment AO in Figure 1;

H is an altitude equal to the segment of OS.

Truncated cone

There is a direct circular cone. If the plane, perpendicular height, cut off the upper part, then the truncated cone is. Two of its bases have a circle shape with radius R 1 and R 2.

If a direct cone is formed by the rotation of the rectangular triangle, then a truncated cone - the rotation of the rectangular trapezium around the straight side.

The volume of a truncated cone is calculated by the following formula:

V \u003d n * (R 1 2 + R 2 2 + R 1 * R 2) * H / 3.

Cone and its cross section plane

Peru ancient Greek mathematics Apollonia Perga owns the theoretical work of "conical sections". Thanks to its work in geometry, curved definitions appeared: parabolas, ellipses, hyperboles. Consider, where is the cone.

Take a direct circular cone. If the plane crosses its perpendicular to the axis, then a circle is formed in the context. When the sequential crosses the cone at an angle to the axis, then the ellipse is obtained in the context.

The secant plane, perpendicular to the base and the parallel axis of the cone, forms a hyperbola on the surface. The plane cutting the cone at an angle to the base and parallel tangent to the cone, creates a curve on the surface called parabola.

The solution of the problem

Even a simple task of how to make a bucket of a certain amount, requires knowledge. For example, you need to calculate the size of the bucket so that it has a volume of 10 liters.

V \u003d 10 l \u003d 10 dm 3;

The cone sweep has the form schematically shown in Figure 3.

L - forming a cone.

To find out the surface area of \u200b\u200bthe bucket, which is calculated by the following formula:

S \u003d n * (R 1 + R 2) * L,

it is necessary to calculate the forming. It is found from the size of the volume V \u003d n * (R 1 2 + R 2 2 + R 1 * R 2) * H / 3.

Hence H \u003d 3V / N * (R 1 2 + R 2 2 + R 1 * R 2).

The truncated cone is formed by the rotation of the rectangular trapezium, in which the lateral side is the forming cone.

L 2 \u003d (R 2- R 1) 2 + H 2.

Now we have all the data to build a bucket drawing.

Why are fire buckets have a cone form?

Who wondered why fire buckets would seem to be a strange conical form? And this is not just like that. It turns out that the conical bucket when steaming the fire has many advantages over the usual having form of a truncated cone.

First, as it turns out, the fire bucket is fillter by water and it does not spill on with carrying. Cone, the volume of which is more than an ordinary bucket, at a time allows you to transfer more water.

Secondly, water from it can be splashing at a greater distance than from the usual bucket.

Thirdly, if the conical bucket is angry with the hands and falls into the fire, then all the water is poured onto the fire focus.

All listed factors allow you to save time - the main factor when steaming fire.

Practical use

Schoolchildren often raises the question of why to learn how to calculate the volume of different geometric bodies, including cone.

And engineers-designers are constantly faced with the need to calculate the volume of the conical parts of the parts of the mechanisms. These are the tips of the drill, parts of turning and milling machines. The form of the cone will allow drills to enter the material, without requiring the initial mark with a special tool.

The volume of the cone has a bunch of sand or land, batted on the ground. If necessary, carrying out simple measurements, it is possible to calculate its volume. Some will cause difficulty as a question of how to find out the radius and height of the heap of sand. Armed with a tape measure, we measure the circumference of the Kholmik C. According to the formula R \u003d C / 2N, we learn the radius. By throwing the rope (roulette) through the vertex, we find the length of the forming. And to calculate the height of the Pythagora theorem and the volume will not be difficult. Of course, this calculation is approximate, but allows you to determine, did not deceive you, bringing a ton of sand instead of Cuba.

Some buildings have the shape of a truncated cone. For example, the Ostankino TV bash is approaching the form of a cone. It can be submitted consisting of two cones supplied to each other. The dome of the vintage locks and cathedrals are a cone, the volume of which ancient architect was calculated with an amazing accuracy.

If you carefully look at the surrounding subjects, then many of them are cones:

• funnels leaks for pouring liquids;
• rule-loudspeaker;
• parking cones;
• lamp shade;
• the familiar Christmas tree;
• wind musical instruments.

As can be seen from the above examples, the ability to calculate the volume of the cone, its surface area is necessary in professional and everyday life. We hope that the article will come to help you.