How to make a scan - a pattern for a cone or a truncated cone of a given size. Simple sweep calculation. The volume of the cone, its calculation The volume of the mustache of the cone

Among the variety of geometric bodies, one of the most interesting is the cone. It is formed by rotating a right triangle around one of its legs.

How to find the volume of a cone - basic concepts

Before you start calculating the volume of a cone, you should familiarize yourself with the basic concepts.

Circular cone - the base of such a cone is a circle. If the base is an ellipse, parabola or hyperbola, then the figures are called elliptical, parabolic or hyperbolic cones. It is worth remembering that the last two types of cones have an infinite volume.
A truncated cone is a part of a cone located between the base and a plane parallel to this base, located between the top and the base.
Height - a segment perpendicular to the base, released from the top.
The generatrix of a cone is a segment that connects the border of the base and the top.

Cone volume

To calculate the volume of a cone, the formula V=1/3*S*H is used, where S is the base area, H is the height. Since the base of the cone is a circle, its area is found by the formula S= nR^2, where n = 3.14, R is the radius of the circle.

There is a situation when some of the parameters are unknown: height, radius or generatrix. In this case, it is worth resorting to the Pythagorean theorem. The axial section of the cone is an isosceles triangle, consisting of two right-angled triangles, where l is the hypotenuse, and H and R are the legs. Then l=(H^2+R^2)^1/2.

Truncated Cone Volume

A truncated cone is a cone with a cut off top.

To find the volume of such a cone, you need the formula:

V=1/3*n*H*(r^2+rR+R^2),

where n=3.14, r is the radius of the section circle, R is the radius of the large base, H is the height.

The axial section of the truncated cone will be an isosceles trapezoid. Therefore, if it is necessary to find the length of the generatrix of a cone or the radius of one of the circles, it is worth using formulas for finding the sides and bases of a trapezoid.

Find the volume of a cone if its height is 8 cm and the base radius is 3 cm.

Given: H=8 cm, R=3 cm.

First, find the area of the base by applying the formula S=nR^2.

S=3.14*3^2=28.26cm^2

Now, using the formula V=1/3*S*H, we find the volume of the cone.

V=1/3*28.26*8=75.36cm^3

Cone-shaped figures are found everywhere: parking cones, building towers, lamp shade. Therefore, knowing how to find the volume of a cone can sometimes come in handy both in professional and everyday life.

Instead of the word “pattern”, “sweep” is sometimes used, but this term is ambiguous: for example, a reamer is a tool for increasing the diameter of a hole, and in electronic technology there is a concept of a reamer. Therefore, although I am obliged to use the words “cone sweep” so that search engines can find this article using them, I will use the word “pattern”.

Building a pattern for a cone is a simple matter. Let us consider two cases: for a full cone and for a truncated one. On the picture (click to enlarge) sketches of such cones and their patterns are shown. (I note right away that we will only talk about straight cones with a round base. We will consider cones with an oval base and inclined cones in the following articles).

1. Full taper

Designations:

Pattern parameters are calculated by the formulas:
;
;
where .

2. Truncated cone

Designations:

Formulas for calculating pattern parameters:
;
;
;
where .
Note that these formulas are also suitable for the full cone if we substitute .

Sometimes, when constructing a cone, the value of the angle at its vertex (or at the imaginary vertex, if the cone is truncated) is fundamental. The simplest example is when you need one cone to fit snugly into another. Let's denote this angle with a letter (see picture).
In this case, we can use it instead of one of the three input values: , or . Why "together about", not "together e"? Because three parameters are enough to construct a cone, and the value of the fourth is calculated through the values of the other three. Why exactly three, and not two or four, is a question that is beyond the scope of this article. A mysterious voice tells me that this is somehow connected with the three-dimensionality of the “cone” object. (Compare with the two initial parameters of the two-dimensional circle segment object, from which we calculated all its other parameters in the article.)

Below are the formulas by which the fourth parameter of the cone is determined when three are given.

4. Methods for constructing a pattern

Calculate the values on the calculator and build a pattern on paper (or immediately on metal) using a compass, ruler and protractor.
Enter formulas and source data into a spreadsheet (for example, Microsoft Excel). The result obtained is used to build a pattern using a graphic editor (for example, CorelDRAW).
use my program, which will draw on the screen and print out a pattern for a cone with the given parameters. This pattern can be saved as a vector file and imported into CorelDRAW.

5. Not parallel bases

As far as truncated cones are concerned, the Cones program still builds patterns for cones that have only parallel bases.
For those who are looking for a way to construct a truncated cone pattern with non-parallel bases, here is a link provided by one of the site visitors:
A truncated cone with non-parallel bases.

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

Sweep construction options:

Development of a right circular cone

The development of the lateral surface of a right circular cone is a circular sector, the radius of which is equal to the length of the generatrix of the conical surface l, and the central angle φ is determined by the formula φ=360*R/l, where R is the radius of the circumference of the base of the cone.

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

Construction algorithm

We inscribe a polygonal pyramid into the conical surface. The more side faces of the inscribed pyramid, the more accurate the correspondence between the actual and approximate scan.
We build a development of the side surface of the pyramid using the triangle method. The points belonging to the base of the cone are connected by a smooth curve.

Example

In the figure below, a regular hexagonal pyramid SABCDEF is inscribed in a right circular cone, and an approximate development of its lateral surface consists of six isosceles triangles - the faces of the pyramid.

Consider a triangle S 0 A 0 B 0 . The lengths of its sides S 0 A 0 and S 0 B 0 are equal to the generatrix l of the conical surface. The value 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 the drawing, we set aside the segment S 0 A 0 =l, after which we draw circles with a radius S 0 B 0 =l and A 0 B 0 = A'B' from points S 0 and A 0 respectively. We connect the point of intersection of 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 of the SABCDEF pyramid are built similarly to the triangle S 0 A 0 B 0 .

Points A, B, C, D, E and F, lying at the base of the cone, are connected by a smooth curve - an arc of a circle, the radius of which is equal to l.

Oblique cone development

Consider the procedure for constructing a sweep of the lateral surface of an inclined cone by the approximation method.

Algorithm

We inscribe the hexagon 123456 in the circle of the base of the cone. Connect the points 1, 2, 3, 4, 5 and 6 with the vertex S. The pyramid S123456 constructed in this way is, with a certain degree of approximation, a replacement for the conical surface and is used as such in further constructions.
We determine the natural values of the edges of the pyramid using the method of rotation around the projecting line: in the example, the i-axis is used, which is perpendicular to the horizontal projection plane and passes through the vertex S.
So, as a result of the rotation of the edge S5, its new horizontal projection S'5' 1 takes a position in which it is parallel to the frontal plane π 2 . Accordingly, S''5'' 1 is the natural value of S5.
We construct a development of the lateral surface of the pyramid S123456, consisting of six triangles: 0 1 0 . The construction of each triangle is performed on three sides. For example, △S 0 1 0 6 0 has the length S 0 1 0 =S''1'' 0 , S 0 6 0 =S''6'' 1 , 1 0 6 0 =1'6'.

The degree of correspondence of the approximate sweep to the actual one depends on the number of faces of the inscribed pyramid. The number of faces is chosen based on the ease of reading the drawing, the requirements for its accuracy, the presence of characteristic points and lines that need to be transferred to the scan.

Transferring a line from the surface of a cone to a development

The line n lying on the surface of the cone is formed as a result of its intersection with a certain plane (figure below). Consider the algorithm for constructing line n on the sweep.

Algorithm

Find the projections of points A, B and C, in which the line n intersects the edges of the pyramid inscribed in the cone S123456.
We determine the actual size of the segments SA, SB, SC by rotating around the projecting line. In this example, SA=S''A'', SB=S''B'' 1 , SC=S''C'' 1 .
We find the position of points A 0 , B 0 , C 0 on the corresponding edges of the pyramid, setting aside segments S 0 A 0 =S''A'', S 0 B 0 =S''B'' 1 , S 0 C 0 =S''C'' 1 .
We connect points A 0 , B 0 , C 0 with a smooth line.

Truncated cone development

The method for constructing a sweep of a right circular truncated cone, described below, is based on the principle of similarity.

In geometry, a truncated cone is a body that is formed by the rotation of a rectangular trapezoid about that side of it, which is perpendicular to the base. How do they calculate truncated cone volume, everyone knows from the school geometry course, and in practice this knowledge is often used by designers of various machines and mechanisms, developers of some consumer goods, as well as architects.

Calculation of the volume of a truncated cone

The formula for calculating the volume of a truncated cone

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

πh (R 2 + R × r + r 2)

h- cone height

r- radius of the upper base

R- bottom base radius

V- volume of the truncated cone

π - 3,14

With such geometric bodies as truncated cones, in everyday life, everyone encounters quite often, if not constantly. Their shape has a wide variety of containers widely used in everyday life: buckets, glasses, some cups. It goes without saying that the designers who developed them must have used a formula that calculates truncated cone volume, since this value is very important in this case, because it determines such an important characteristic as the capacity of the product.

Engineering structures, which are truncated cones, can often be seen at large industrial enterprises, as well as thermal and nuclear power plants. It is this form that cooling towers have - devices designed to cool large volumes of water by forcing a counter flow of atmospheric air. Most often, these designs are used in cases where it is required to significantly reduce the temperature of a large amount of liquid in a short time. The developers of these structures must determine truncated cone volume the formula for calculating which is quite simple and known to all those who once studied well in high school.

Details having this geometric shape are quite often found in the design of various technical devices. For example, gears used in systems where it is required to change the direction of kinetic transmission are most often implemented using bevel gears. These parts are an integral part of a wide variety of gearboxes, as well as automatic and manual gearboxes used in modern vehicles.

The shape of a truncated cone has some cutting tools that are widely used in production, for example, milling cutters. With their help, you can process inclined surfaces at a certain angle. For sharpening cutters of metalworking and woodworking equipment, abrasive wheels are often used, which are also truncated cones. Besides, truncated cone volume it is required to determine the designers of turning and milling machines, which involve the fastening of a cutting tool equipped with tapered shanks (drills, reamers, etc.).