Conical cone surface. Cone concept. The concept of a generatrix of a cone
Definition. Cone apex is the point (K) from which the rays emanate.
Definition. Base of the cone is the plane formed by the intersection of a flat surface and all rays emanating from the top of the cone. A cone can have stems such as a circle, ellipse, hyperbola, and parabola.
Definition. Generatrix of the cone (L) is any line segment that connects the top of the cone with the boundary of the base of the cone. The generator is a segment of the ray emerging from the top of the cone.
Formula. Length of generatrix (L) a straight circular cone through the radius R and the height H (through the Pythagorean theorem):
Definition. Guide A cone is a curve that describes the contour of the base of the cone.
Definition. Side surface a cone is a collection of all generators of a cone. That is, the surface that is formed by the movement of the generatrix along the guide of the cone.
Definition. Surface The cone consists of a lateral surface and a base of the cone.
Definition. Height cone (H) is a line segment that extends from the top of the cone and is perpendicular to its base.
Definition. Axis cone (a) is a straight line passing through the top of the cone and the center of the base of the cone.
Definition. Taper (C) cone is the ratio of the diameter of the base of the cone to its height. In the case of a truncated cone, this is the ratio of the difference between the diameters of the cross sections D and d of the truncated cone to the distance between them: where R is the base radius, and H is the height of the cone.
In mechanical engineering, along with cylindrical ones, parts with conical surfaces in the form of outer cones or in the form of conical holes are widely used. For example, the center of a lathe has two outer cones, one of which is used to install and fix it in the tapered bore of the spindle; an external cone for installation and fastening also have a drill, countersink, reamer, etc. The adapter sleeve for fastening drills with a tapered shank has an external cone and a tapered hole
1. The concept of a cone and its elements
Cone elements. If you rotate the right-angled triangle ABC around the leg AB (Fig. 202, a), then the AVG body is formed, called full cone... The AB line is called the axis or cone height, line AB - generatrix of the cone... Point A is the top of the cone.
When the BV leg rotates around the AB axis, a circle surface is formed, called base of the cone.
The angle of VAG between the lateral sides AB and AG is called taper angle and is denoted by 2α. The half of this angle formed by the side of the AG and the AB axis is called taper angle and is denoted by α. Angles are expressed in degrees, minutes and seconds.
If we cut off its upper part from a full cone with a plane parallel to its base (Fig. 202, b), we get a body called truncated cone... It has two bases, upper and lower. The distance OO 1 along the axis between the bases is called truncated cone height... Since in mechanical engineering for the most part it is necessary to deal with parts of the cones, that is, truncated cones, they are usually simply called cones; in what follows we will call all conical surfaces cones.
The relationship between the elements of the cone. The drawing usually indicates three main dimensions of the cone: the larger diameter D, the smaller one - d and the height of the cone l (Fig. 203). 
Sometimes only one of the diameters of the cone is indicated in the drawing, for example, the larger D, the height of the cone l and the so-called taper. Taper is the ratio of the difference between the diameters of the cone to its length. We denote the taper by the letter K, then
If the cone has dimensions: D \u003d 80 mm, d \u003d 70 mm and l \u003d 100 mm, then according to the formula (10):
This means that over a length of 10 mm, the diameter of the cone decreases by 1 mm, or for every millimeter of the length of the cone, the difference between its diameters changes by
Sometimes in the drawing, instead of the angle of the cone, it is indicated cone slope... The slope of the cone shows to what extent the generatrix of the cone deviates from its axis.
The slope of the cone is determined by the formula
where tg α is the slope of the cone;
l - cone height in mm.
Using formula (11), you can use trigonometric tables to determine the angle a of the slope of the cone.
Example 6. Given D \u003d 80 mm; d \u003d 70mm; l \u003d 100 mm. According to the formula (11) we have According to the table of tangents we find the value closest to tan α \u003d 0.05, i.e., tan α \u003d 0.049, which corresponds to the slope angle of the cone α \u003d 2 ° 50 ". Consequently, the angle of the cone 2α \u003d 2 · 2 ° 50 "\u003d 5 ° 40".
The taper slope and taper are usually expressed in simple fractions, for example: 1: 10; 1: 50, or a decimal fraction, for example, 0.1; 0.05; 0.02, etc.
2. Methods for obtaining tapered surfaces on a lathe
On a lathe, conical surfaces are processed in one of the following ways:
a) by turning the upper part of the caliper;
b) lateral displacement of the tailstock body;
c) using a tapered ruler;
d) using a wide incisor.
3. Processing of tapered surfaces by turning the upper part of the caliper
When making on a lathe short outer and inner conical surfaces with a large slope angle, you need to turn the upper part of the support relative to the machine axis at an angle α of the slope of the cone (see Fig. 204). With this method of work, the feed can only be made by hand, by rotating the handle of the lead screw of the upper part of the caliper, and only in the most modern lathes there is a mechanical feed of the upper part of the caliper.
To install the upper part of the support 1 at the required angle, you can use the graduations marked on the flange 2 of the rotary part of the support (Fig. 204). If the angle α of the taper of the cone is given according to the drawing, then the upper part of the support is rotated together with its rotary part by the required number of divisions denoting degrees. The number of divisions is counted relative to the marks marked on the bottom of the caliper.
If the angle α is not given in the drawing, but the larger and smaller diameters of the cone and the length of its conical part are indicated, then the value of the angle of rotation of the support is determined by the formula (11)
Example 7. Given the diameters of the cone D \u003d 80 mm, d \u003d 66 mm, the length of the cone l \u003d 112 mm. We have: 
According to the table of tangents we find approximately: a \u003d 3 ° 35 ". Therefore, the upper part of the support must be turned by 3 ° 35".
The method of turning the conical surfaces by turning the upper part of the support has the following disadvantages: it usually allows the use of only manual feed, which affects labor productivity and cleanliness of the treated surface; allows turning relatively short tapered surfaces limited by the stroke length of the upper part of the caliper.
4. Processing of tapered surfaces by the method of lateral displacement of the tailstock body
To obtain a conical surface on a lathe, it is necessary to move the tip of the cutter not parallel, but at a certain angle to the center axis when rotating the workpiece. This angle must be equal to the angle α of the taper of the cone. The easiest way to get the angle between the centerline and feed direction is to offset the centerline by moving the trailing center laterally. By shifting the rear center towards the cutter (towards itself), as a result of turning, a cone is obtained, in which the larger base is directed towards the headstock; when the rear center is displaced in the opposite direction, that is, from the cutter (away from you), the larger base of the cone will be on the side of the tailstock (Fig. 205).
The displacement of the tailstock body is determined by the formula
where S is the displacement of the tailstock body from the headstock spindle axis in mm;
D is the diameter of the large base of the cone in mm;
d is the diameter of the small base of the cone in mm;
L is the length of the entire part or the distance between centers in mm;
l is the length of the tapered part of the part in mm.
Example 8. Determine the offset of the tailstock center for turning a truncated cone, if D \u003d 100 mm, d \u003d 80 mm, L \u003d 300 mm and l \u003d 200mm. By formula (12) we find:
The displacement of the tailstock body is made using divisions 1 (Figure 206), marked on the end of the base plate, and at risk 2 at the end of the tailstock body.
If there are no divisions at the end of the plate, then the tailstock housing is shifted using a measuring ruler, as shown in Fig. 207.
The advantage of machining tapered surfaces by offsetting the tailstock body is that long taper lengths can be turned in this way and can be turned with power feed.
Disadvantages of this method: inability to bore tapered holes; loss of time to rearrange the tailstock; the ability to handle only gentle cones; misalignment of the centers in the center holes, which leads to quick and uneven wear of the centers and center holes and causes rejects when the part is re-installed in the same center holes.
Uneven wear of the center holes can be avoided by using a special ball center instead of the usual one (Fig. 208). Such centers are used primarily for the processing of precise tapers.
5. Processing of tapered surfaces using a tapered ruler
For processing tapered surfaces with a slope angle of up to 10-12 °, modern lathes usually have a special device called a tapered ruler. The cone processing scheme using a tapered ruler is shown in Fig. 209.
A plate 11 is attached to the machine bed, on which a tapered ruler 9 is installed. The ruler can be rotated around the pin 8 at the required angle a to the axis of the workpiece. To fix the ruler in the required position, there are two bolts 4 and 10. A slider 7 freely slides along the ruler, which connects to the lower transverse part 12 of the support using a rod 5 and a clamp 6. So that this part of the support can freely slide along the guides, it is disconnected from the carriage 3 by unscrewing the cross screw or disconnecting the nut from the caliper.
If you tell the carriage a longitudinal feed, then the slider 7, captured by the rod 5, will begin to move along the ruler 9. Since the slider is fastened to the cross slide of the slide, they, together with the cutter, will move parallel to the ruler 9. Due to this, the cutter will machine a tapered surface with a slope angle equal to the angle α of rotation of the tapered ruler.
After each pass, the cutter is set to the cutting depth using the handle 1 of the upper part 2 of the support. This part of the caliper must be rotated 90 ° relative to the normal position, i.e. as shown in fig. 209.
If the diameters of the bases of the cone D and d and its length l are given, then the angle of rotation of the ruler can be found by formula (11).
Having calculated the value of tg α, it is easy to determine the value of the angle α from the table of tangents.
The use of a tapered ruler has several advantages:
1) adjusting the ruler is convenient and quick;
2) when switching to the processing of cones, it is not required to disrupt the normal adjustment of the machine, that is, it is not necessary to displace the body of the tailstock; the centers of the machine remain in the normal position, that is, on one axis, due to which the center holes in the parts and the centers of the machine are not triggered;
3) using a tapered ruler, you can not only grind the outer tapered surfaces, but also bore tapered holes;
4) it is possible to work with a longitudinal self-propelled gun, which increases labor productivity and improves the quality of processing.
The disadvantage of a tapered rule is the need to disconnect the slide slide from the cross feed screw. This drawback is eliminated in the design of some lathes, in which the screw is not rigidly connected to its handwheel and toothed wheels of the transverse self-propelled.
6. Processing of tapered surfaces with a wide cutter
The processing of tapered surfaces (external and internal) with a small length of the cone can be done with a wide cutter with an angle in the plan corresponding to the angle α of the slope of the cone (Fig. 210). The cutter feed can be longitudinal and transverse.
However, the use of a wide cutter on conventional machines is only possible with a cone length not exceeding about 20 mm. It is possible to use wider cutters only on particularly rigid machines and parts, if this does not cause vibration of the cutter and the workpiece.
7. Boring and reaming of tapered holes
Tapered hole machining is one of the most difficult turning jobs; it is much more difficult than machining the outer tapers.
The processing of tapered holes on lathes in most cases is carried out by boring with a cutter with a turn of the upper part of the support and less often using a tapered ruler. All calculations associated with turning the upper part of the caliper or tapered ruler are performed in the same way as when turning the outer tapered surfaces.
If the hole must be in solid material, then first a cylindrical hole is drilled, which is then bored with a taper cutter or processed with conical countersinks and reamers.
To speed up boring or reaming, you should first drill a hole with a drill, diameter d, which is 1-2 mm less than the diameter of the small base of the cone (Fig. 211, a). After that, the hole is drilled with one (Fig. 211, b) or two (Fig. 211, c) drills to obtain steps.
After finishing boring of the cone, it is deployed with a conical sweep of the appropriate taper. For tapers with a small taper, it is more profitable to process tapered holes directly after drilling with a set of special reamers, as shown in Fig. 212.
8. Cutting conditions when machining holes with conical reamers
Tapered reamers work in more severe conditions than cylindrical reamers: while cylindrical reamers remove a small allowance with small cutting edges, tapered reamers cut the entire length of their cutting edges located on the generatrix of the cone. Therefore, when working with conical reamers, feed rates and cutting speeds are used less than when working with cylindrical reamers.
When machining holes with conical reamers, the feed is done manually by rotating the tailstock handwheel. Make sure that the tailstock quill moves evenly.
Feeds when unrolling steel 0.1-0.2 mm / rev, while unrolling cast iron 0.2-0.4 mm / rev.
Cutting speed when reaming tapered holes with reamers from high speed steel 6-10 m / min.
Cooling should be used to facilitate the operation of the conical reamers and to obtain a clean and smooth surface. When processing steel and cast iron, an emulsion or sulfofresol is used.
9. Measuring tapered surfaces
The surfaces of the cones are checked with templates and gauges; measurement and at the same time check of the angles of the cone is carried out by protractors. In fig. 213 shows a method for checking a cone using a template.
The outer and inner angles of various parts can be measured with a universal goniometer (Fig. 214). It consists of a base 1, on which the main scale is marked on the arc 130. A ruler 5 is rigidly attached to the base 1. Sector 4, which carries the vernier 3, moves along the base arc. the ability to move along the edge of sector 4.
By means of various combinations in the installation of the measuring parts of the protractor, it is possible to measure angles from 0 to 320 °. The reading value for the vernier is 2 ". The reading obtained when measuring the angles is made according to the scale and the vernier (Fig. 215) as follows: the zero stroke of the vernier indicates the number of degrees, and the stroke of the vernier, which coincides with the stroke of the base scale, indicates the number of minutes. 215 with the stroke of the base scale coincides with the 11th stroke of the vernier, which means 2 "X 11 \u003d 22". Therefore, the angle in this case is equal to 76 ° 22 ".
In fig. 216 shows combinations of measuring parts of a universal protractor, which allow measurement of various angles from 0 to 320 °.
For a more accurate check of the cones in mass production, special gauges are used. In fig. 217, a shows a tapered bushing gauge for checking the outer cones, and in Fig. 217, b-taper gauge-plug for testing tapered holes.
On the gauges, ledges 1 and 2 are made at the ends or risks 3 are applied, which serve to determine the accuracy of the surfaces to be checked.
On. fig. 218 shows an example of checking a tapered bore with a plug gauge.
To check the hole, the gauge (see Fig. 218), having a ledge 1 at a certain distance from the end 2 and two risks 3, is inserted with light pressure into the hole and check if the gauge is swinging in the hole. No wobble indicates that the taper angle is correct. After making sure that the angle of the cone is correct, they begin to check its size. To do this, observe to what place the caliber will enter the checked part. If the end of the part cone coincides with the left end of the step 1 or with one of the notches 3 or is between the risks, then the dimensions of the cone are correct. But it may happen that the gauge enters the part so deeply that both risks 3 enter the hole or both ends of the ledge 1 come out of it. This shows that the hole diameter is larger than the specified one. If, on the contrary, both risks are outside the hole or none of the ends of the ledge comes out of it, then the hole diameter is less than required.
The following method is used to accurately check the taper. On the measured surface of the part or gauge, two or three lines are drawn with chalk or a pencil along the generatrix of the cone, then the gauge is inserted or put on the part and turned by a part of a turn. If the lines are erased unevenly, this means that the taper of the part is not machined accurately and needs to be corrected. Erasing lines at the ends of the gauge indicates incorrect taper; the erasure of the lines in the middle of the gauge shows that the cone has a slight concavity, which is usually caused by the inaccurate position of the tip of the cutter in the height of the centers. Instead of chalk lines, you can apply a thin layer of special paint (blue) to the entire conical surface of a part or caliber. This method provides greater measurement accuracy.
10. Defect in the processing of tapered surfaces and measures to prevent it
When processing tapered surfaces, in addition to the mentioned types of scrap for cylindrical surfaces, the following types of scrap are additionally possible:
1) incorrect taper;
2) deviations in the size of the cone;
3) deviations in the dimensions of the diameters of the bases with the correct taper;
4) non-straightness of the generatrix of the conical surface.
1. Incorrect taper is mainly due to inaccurate displacement of the tailstock body, inaccurate rotation of the upper part of the caliper, improper installation of tapered rule, improper sharpening or installation of a wide cutter. Consequently, by accurately setting the tailstock body, the upper part of the caliper or the tapered ruler before starting processing, defects can be prevented. We can correct this type of marriage only if the error in the entire length of the cone is directed into the body of the part, i.e., all diameters of the sleeve are smaller, and that of the conical rod are larger than required.
2. The wrong size of the cone at the correct angle, ie, the wrong size of the diameters along the entire length of the cone, is obtained if not enough or too much material has been removed. Defects can be prevented only by carefully setting the cutting depth along the dial on finishing passes. We will fix the marriage if not enough material has been removed.
3. It can happen that with the correct taper and exact dimensions of one end of the cone, the diameter of the other end is incorrect. The only reason is non-compliance with the required length of the entire tapered section of the part. We will fix the marriage if the part is too long. To avoid this type of scrap, it is necessary to carefully check its length before processing the cone.
4. The non-straightness of the generatrix of the cone being machined is obtained when the cutter is installed above (Fig. 219, b) or below (Fig. 219, c) the center (in these figures, for greater clarity, the distortions of the generatrix of the cone are shown in a highly exaggerated form). Thus, this type of marriage is the result of the turner's careless work.
test questions 1. What methods can be used to machine tapered surfaces on lathes?
2. When is it recommended to turn the upper part of the caliper?
3. How is the angle of rotation of the upper part of the taper turning caliper calculated?
4. How is the correct rotation of the upper part of the caliper checked?
5. How to check the tailstock housing offset? How to calculate the offset amount?
6. What are the main elements of a tapered ruler? How to adjust a tapered ruler for a given part?
7. Set the following angles on the universal goniometer: 50 ° 25 "; 45 ° 50"; 75 ° 35 ".
8. What instruments are used to measure tapered surfaces?
9. Why are ledges or marks made on tapered gauges and how to use them?
10. List the types of rejects when processing tapered surfaces. How to avoid them?
Which emanates from one point (the top of the cone) and which pass through a flat surface.
It happens that a cone is a part of a body that has a limited volume and which is obtained by combining each segment that connects the vertex and points of a flat surface. The latter, in this case, is base of the cone, and the cone is called resting on a given base.
When the base of the cone is a polygon, this is already pyramid .
|
Circular cone is a body consisting of a circle (the base of the cone), a point that does not lie in the plane of this circle (the top of the cone and all the segments that connect the top of the cone with the base points). The segments that connect the top of the cone and the points of the circumference of the base are called generators of the cone... The cone surface consists of a base and a lateral surface. |
The lateral surface area is correct n-gonal pyramid inscribed in a cone:
S n \u003d ½P n l n,
where P n is the perimeter of the base of the pyramid, and l n - apothem.
By the same principle: for the lateral surface area of \u200b\u200ba truncated cone with base radii R 1, R 2 and generating l we get the following formula:
S \u003d (R 1 + R 2) l.
Straight and oblique circular cones with equal base and height. These bodies have the same volume:
Cone properties.
- When the area of \u200b\u200bthe base has a limit, then the volume of the cone also has a limit and is equal to the third part of the product of the height and the area of \u200b\u200bthe base.
where S - base area, H - height.
Thus, each cone that rests on this base and has a vertex that is located on a plane parallel to the base has an equal volume, since their heights are the same.
- The center of gravity of each cone with a limit volume is one quarter of the height from the base.
- The solid angle at the apex of a right circular cone can be expressed by the following formula:
where α - cone opening angle.
- The lateral surface area of \u200b\u200bsuch a cone, formula:
and the total surface area (that is, the sum of the lateral and base areas), the formula is:
S \u003d πR (l + R),
where R - base radius, l- generatrix length.
- Volume of a circular cone, formula:
- For a truncated cone (not just straight or circular) volume, the formula is:
where S 1 and S 2 - the area of \u200b\u200bthe upper and lower bases,
h and H - the distance from the plane of the upper and lower base to the top.
- The intersection of a plane with a right circular cone is one of the conic sections.
) is a body in Euclidean space obtained by combining all rays emanating from one point ( tops cone) and passing through a flat surface. Sometimes a cone is called a part of such a body, which has a limited volume and is obtained by combining all the segments connecting the vertex and points of a flat surface (the latter in this case is called basis cone, and the cone is called leaning on this basis). If the base of the cone is a polygon, that cone is a pyramid.
Encyclopedic YouTube
-
Subtitles
1 / 4
✪ How to make a paper cone.
Related definitions
- The segment connecting the top and the base boundary is called generatrix of the cone.
- The union of the generators of the cone is called generatrix (or side) cone surface... The forming surface of the cone is a conical surface.
- A segment lowered perpendicularly from the vertex to the base plane (as well as the length of such a segment) is called cone height.
- Cone opening angle - the angle between two opposite generatrices (angle at the top of the cone, inside the cone).
- If the base of the cone has a center of symmetry (for example, it is a circle or an ellipse) and the orthogonal projection of the apex of the cone onto the plane of the base coincides with this center, then the cone is called direct... In this case, the straight line connecting the top and the center of the base is called axis of the cone.
- Oblique (inclined) cone - a cone whose orthogonal projection of the vertex to the base does not coincide with its center of symmetry.
- Circular cone - a cone whose base is a circle.
- Straight circular cone (often simply called a cone) can be obtained by rotating a right-angled triangle around a straight line containing a leg (this straight line is the axis of the cone).
- A cone resting on an ellipse, parabola or hyperbola is called, respectively elliptical, parabolic and hyperbolic cone (the last two have infinite volume).
- The part of the cone lying between the base and a plane parallel to the base and located between the top and the base is called truncated cone, or conical layer.
Properties
- If the area of \u200b\u200bthe base is finite, then the volume of the cone is also finite and is equal to a third of the product of the height and the area of \u200b\u200bthe base.
where S - base area, H - height. Thus, all cones resting on a given base (of finite area) and having a vertex located on a given plane parallel to the base have the same volume, since their heights are equal.
- The center of gravity of any cone with a finite volume lies at a quarter of the height from the base.
- The solid angle at the apex of a right circular cone is
- The lateral surface area of \u200b\u200bsuch a cone is
and the total surface area (that is, the sum of the lateral and base areas)
S \u003d π R (l + R), (\\ displaystyle S \u003d \\ pi R (l + R),) Where R - base radius, l \u003d R 2 + H 2 (\\ displaystyle l \u003d (\\ sqrt (R ^ (2) + H ^ (2)))) - generatrix length.- The volume of a circular (not necessarily straight) cone is
- For a truncated cone (not necessarily straight and circular), the volume is:
where S 1 and S 2 are the areas of the upper (closest to the top) and lower bases, respectively, h and H - the distance from the plane, respectively, of the upper and lower base to the top.
- The intersection of a plane with a right circular cone is one of the conic sections (in non-degenerate cases - an ellipse, parabola or hyperbola, depending on the position of the secant plane).
Cone Equation
Equations defining the lateral surface of a straight circular cone with an opening angle of 2Θ, apex at the origin and an axis coinciding with the axis Oz :
- In a spherical coordinate system with coordinates ( r, φ, θ) :
- In a cylindrical coordinate system with coordinates ( r, φ, z) :
- In a Cartesian coordinate system with coordinates (x, y, z) :
where the constants a, with determined by the proportion c / a \u003d cos \u2061 Θ / sin \u2061 Θ. (\\ displaystyle c / a \u003d \\ cos \\ Theta / \\ sin \\ Theta.) Hence, it can be seen that the lateral surface of a straight circular cone is a second-order surface (it is called conical surface). In general, the second-order conical surface rests on an ellipse; in a suitable Cartesian coordinate system (axes Oh and OU are parallel to the axes of the ellipse, the vertex of the cone coincides with the origin, the center of the ellipse lies on the axis Oz ) its equation has the form
x 2 a 2 + y 2 b 2 - z 2 c 2 \u003d 0, (\\ displaystyle (\\ frac (x ^ (2)) (a ^ (2))) + (\\ frac (y ^ (2)) ( b ^ (2))) - (\\ frac (z ^ (2)) (c ^ (2))) \u003d 0,)moreover a / c and b / c are equal to the semiaxes of the ellipse. In the most general case, when the cone rests on an arbitrary flat surface, it can be shown that the equation for the lateral surface of the cone (with apex at the origin) is given by the equation f (x, y, z) \u003d 0, (\\ displaystyle f (x, y, z) \u003d 0,) where function f (x, y, z) (\\ displaystyle f (x, y, z)) is homogeneous, that is, satisfying the condition f (α x, α y, α z) \u003d α nf (x, y, z) (\\ displaystyle f (\\ alpha x, \\ alpha y, \\ alpha z) \u003d \\ alpha ^ (n) f (x, y , z)) for any real number α.
Scan
A straight circular cone as a body of revolution is formed by a right-angled triangle rotating around one of the legs, where h - the height of the cone from the center of the base to the top - is the leg of a right-angled triangle around which rotation occurs. Second leg of a right triangle r is the radius at the base of the cone. The hypotenuse of a right triangle is l - generatrix of the cone.
Only two quantities can be used to create a flat pattern. r and l ... Base radius r defines the circle of the base of the cone in the sweep, and the sector of the lateral surface of the cone defines the generator of the lateral surface l , which is the radius of the side surface sector. Sector angle φ (\\ displaystyle \\ varphi) in the sweep of the lateral surface of the cone is determined by the formula:
φ \u003d 360 ° ( r/l) .Today we will tell you how to find the generatrix of a cone, which is often required in school geometry problems.
The concept of a generatrix of a cone
A straight cone is a shape that is obtained by rotating a right-angled triangle around one of its legs. The base of the cone forms a circle. The vertical section of the cone is a triangle, the horizontal section is a circle. The height of the cone is the line segment connecting the top of the cone to the center of the base. The generatrix of a cone is a line segment that connects the top of the cone to any point on the base circumference line.
Since the cone is formed by the rotation of a right-angled triangle, it turns out that the first leg of such a triangle is the height, the second is the radius of the circle lying at the base, and the generatrix of the cone will be the hypotenuse. It is easy to guess that the Pythagorean theorem is useful for calculating the length of the generator. And now more about how to find the length of the generatrix of the cone.
Find the generator
The easiest way to understand how to find a generatrix is \u200b\u200bwith a specific example. Suppose the following conditions of the problem are given: the height is 9 cm, the diameter of the base circle is 18 cm. It is necessary to find the generator.
So, the height of the cone (9 cm) is one of the legs of the right-angled triangle, with which this cone was formed. The second leg will be the radius of the base circle. The radius is half the diameter. Thus, we divide the given diameter in half and we get the length of the radius: 18: 2 \u003d 9. The radius is 9.
Now it is very easy to find the generatrix of the cone. Since it is a hypotenuse, the square of its length will be equal to the sum of the squares of the legs, that is, the sum of the squares of the radius and height. So, the square of the length of the generatrix \u003d 64 (the square of the length of the radius) + 64 (the square of the length of the height) \u003d 64x2 \u003d 128. Now we extract the square root of 128. As a result, we get eight roots of two. This will be the generatrix of the cone.
As you can see, there is nothing complicated about it. As an example, we took simple conditions of the problem, but in a school course they can be more complicated. Remember that to calculate the length of the generatrix you need to find out the radius of the circle and the height of the cone. Knowing these data, it is easy to find the length of the generatrix.
Essay on the topic: The meaning of the title of Turgenev's novel "Fathers and Sons" (2 options)
The originality of the poem mtsyri as romantic
Pushkin "Eugene Onegin")