Elements of a cone and their definitions. Cone as a geometric figure
Definitions:
Definition 1. Cone
Definition 2. Circular cone
Definition 3. Cone height
Definition 4. Straight cone
Definition 5. Right circular cone
Theorem 1. Generators of the cone
Theorem 1.1. Axial section of the cone
Volume and area:
Theorem 2. Volume of a cone
Theorem 3. Area of the lateral surface of a cone
Frustum :
Theorem 4. Section parallel to the base
Definition 6. Truncated cone
Theorem 5. Volume of a truncated cone
Theorem 6. Lateral surface area of a truncated cone
Definitions
A body bounded on the sides by a conical surface taken between its top and the plane of the guide, and the flat base of the guide formed by a closed curve, is called a cone.
Basic Concepts
A circular cone is a body that consists of a circle (base), a point not lying in the plane of the base (vertex) and all segments connecting the vertex to the points of the base.
A straight cone is a cone whose height contains the center of the base of the cone.
Consider any line (curve, broken or mixed) (for example, l), lying in a certain plane, and an arbitrary point (for example, M) not lying in this plane. All possible straight lines connecting point M to all points of a given line l, form surface called canonical. Point M is the vertex of such a surface, and the given line l - guide. All straight lines connecting point M to all points of the line l, called forming. A canonical surface is not limited by either its vertex or its guide. It extends indefinitely in both directions from the top. Let now the guide be a closed convex line. If the guide is a broken line, then the body, bounded on the sides by a canonical surface taken between its top and the plane of the guide, and a flat base in the plane of the guide, is called a pyramid.
If the guide is a curved or mixed line, then the body bounded on the sides by a canonical surface taken between its top and the plane of the guide, and a flat base in the plane of the guide, is called a cone or
Definition 1
. A cone is a body consisting of a base - a flat figure bounded by a closed line (curved or mixed), a vertex - a point that does not lie in the plane of the base, and all segments connecting the vertex with all possible points of the base.
All straight lines passing through the vertex of the cone and any of the points of the curve bounding the figure of the base of the cone are called generators of the cone. Most often in geometric problems, the generatrix of a straight line means a segment of this straight line, enclosed between the vertex and the plane of the base of the cone.
The base of a limited mixed line is a very rare case. It is indicated here only because it can be considered in geometry. The case with a curved guide is more often considered. Although, both the case with an arbitrary curve and the case with a mixed guideline are of little use and it is difficult to derive any patterns from them. Among the cones, the right circular cone is studied in the course of elementary geometry.
It is known that a circle is a special case of a closed curved line. A circle is a flat figure bounded by a circle. Taking the circle as a guide, we can define a circular cone.
Definition 2
. A circular cone is a body that consists of a circle (base), a point not lying in the plane of the base (vertex) and all segments connecting the vertex to the points of the base.
Definition 3
. The height of a cone is the perpendicular descended from the top to the plane of the base of the cone. You can select a cone, the height of which falls at the center of the flat figure of the base.
Definition 4
. A straight cone is a cone whose height contains the center of the base of the cone.
If we combine these two definitions, we get a cone, the base of which is a circle, and the height falls at the center of this circle.
Definition 5
. A right circular cone is a cone whose base is a circle, and its height connects the top and the center of the base of this cone. Such a cone is obtained by rotating a right triangle around one of its legs. Therefore, a right circular cone is a body of revolution and is also called a cone of revolution. Unless otherwise stated, for brevity in what follows we simply say cone.
So here are some properties of the cone:
Theorem 1.
All generators of the cone are equal. Proof. The height of the MO is perpendicular to all straight lines of the base, by definition, a straight line perpendicular to the plane. Therefore, the triangles MOA, MOB and MOS are rectangular and equal on two legs (MO is the general one, OA=OB=OS are the radii of the base. Therefore, the hypotenuses, i.e., the generators, are also equal.
The radius of the base of the cone is sometimes called cone radius. The height of the cone is also called cone axis, therefore any section passing through the height is called axial section. Any axial section intersects the base in diameter (since the straight line along which the axial section and the plane of the base intersect passes through the center of the circle) and forms an isosceles triangle.
Theorem 1.1.
The axial section of the cone is an isosceles triangle. So triangle AMB is isosceles, because its two sides MB and MA are generators. Angle AMB is the angle at the vertex of the axial section.
Which emanate 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 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 said to rest on this base.
When the base of a cone is a polygon, it is already pyramid .
Circular cone- this 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 segments that connect the top of the cone with the points of the base). The segments that connect the vertex of the cone and the points of the base circle are called forming a cone. The surface of the cone consists of a base and a side surface. |
The lateral surface area is correct n-a carbon pyramid inscribed in a cone:
S n =½P n l n,
Where Pn- the perimeter of the base of the pyramid, and l n- apothem.
By the same principle: for the lateral surface area of a truncated cone with base radii R 1, R 2 and forming l we get the following formula:
S=(R 1 +R 2)l.
Straight and oblique circular cones with equal base and height. These bodies have the same volume:
Properties of a cone.
- When the area of the base has a limit, it means that 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 the 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 equal volume, since their heights are the same.
- The center of gravity of each cone with a volume having a limit is located at a quarter of the height from the base.
- The solid angle at the vertex of a right circular cone can be expressed by the following formula:
Where α - cone opening angle.
- The lateral surface area of such a cone, formula:
and the total surface area (that is, the sum of the areas of the lateral surface and base), the formula:
S=πR(l+R),
Where R- radius of the base, l— length of the generatrix.
- Volume of a circular cone, formula:
- For a truncated cone (not just straight or circular), volume, formula:
Where S 1 And S 2- area of the upper and lower bases,
h And H- distances 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.
In mechanical engineering, along with cylindrical ones, parts with conical surfaces in the form of external 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 serves to install and secure it in the conical hole of the spindle; a drill, countersink, reamer, etc. also have an outer cone for installation and fastening. The adapter sleeve for fastening drills with a conical shank has an outer cone and a conical hole
1. The concept of a cone and its elements
Elements of a cone. If you rotate the right triangle ABC around the leg AB (Fig. 202, a), then a body ABG is formed, called full cone. Line AB is called the axis or cone height, line AB - generatrix of the cone. Point A is the top of the cone.
When the leg BV rotates around the axis AB, a circle surface is formed, called base of the cone.
The angle VAG between the lateral sides AB and AG is called cone angle and is denoted by 2α. Half of this angle formed by the lateral side AG and the axis AB is called cone angle and is denoted by α. Angles are expressed in degrees, minutes and seconds.
If we cut off its upper part from a complete cone with a plane parallel to its base (Fig. 202, b), we obtain 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 we mostly have to deal with parts of cones, i.e. truncated cones, they are usually simply called cones; From now on we will call all conical surfaces cones.
The connection between the elements of the cone. The drawing usually indicates three main dimensions of the cone: the larger diameter D, the smaller diameter d and the height of the cone l (Fig. 203).
Sometimes the drawing indicates only one of the cone diameters, for example, the larger D, the cone height l and the so-called taper. Taper is the ratio of the difference between the diameters of a cone and its length. Let us denote the taper by the letter K, then
If the cone has dimensions: D = 80 mm, d = 70 mm and l = 100 mm, then according to 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 the extent to which the generatrix of the cone deviates from its axis.
The slope of the cone is determined by the formula
where tan α is the slope of the cone;
l is the height of the cone in mm.
Using formula (11), you can use trigonometric tables to determine the angle a of the cone.
Example 6. Given D = 80 mm; d=70mm; l= 100 mm. Using formula (11), we have From the table of tangents we find the value closest to tan α = 0.05, i.e. tan α = 0.049, which corresponds to the cone slope angle α = 2°50". Therefore, the cone angle 2α = 2 ·2°50" = 5°40".
Cone slope and taper are usually expressed as a simple fraction, for example: 1:10; 1:50, or a decimal fraction, for example, 0.1; 0.05; 0.02, etc.
2. Methods for producing conical surfaces on a lathe
On a lathe, conical surfaces are processed in one of the following ways:
a) turning the upper part of the caliper;
b) transverse displacement of the tailstock body;
c) using a cone ruler;
d) using a wide cutter.
3. Machining conical surfaces by turning the upper part of the caliper
When making short external and internal conical surfaces with a large slope angle on a lathe, you need to rotate the upper part of the support relative to the axis of the machine at an angle α of the cone slope (see Fig. 204). With this method of operation, feeding can only be done by hand, rotating the handle of the lead screw of the upper part of the support, and only the most modern lathes have a mechanical feed of the upper part of the support.
To set the upper part of the caliper 1 to the required angle, you can use the divisions marked on the flange 2 of the rotating part of the caliper (Fig. 204). If the slope angle α of the cone is specified according to the drawing, then the upper part of the caliper is rotated together with its rotating part by the required number of divisions indicating degrees. The number of divisions is counted relative to the mark 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 caliper rotation angle is determined by formula (11)
Example 7. The given cone diameters are D = 80 mm, d = 66 mm, cone length l = 112 mm. We have: Using the table of tangents we find approximately: a = 3°35". Therefore, the upper part of the caliper must be rotated 3°35".
The method of turning conical surfaces by turning the upper part of the caliper has the following disadvantages: it usually allows the use of only manual feed, which affects labor productivity and the cleanliness of the machined surface; allows you to grind relatively short conical surfaces limited by the stroke length of the upper part of the caliper.
4. Machining of conical surfaces using the method of transverse displacement of the tailstock body
To obtain a conical surface on a lathe, when rotating the workpiece, it is necessary to move the tip of the cutter not parallel, but at a certain angle to the axis of the centers. This angle must be equal to the slope angle α of the cone. The simplest way to obtain the angle between the center axis and the feed direction is to shift the center line by moving the back center in the transverse direction. By shifting the rear center towards the cutter (toward itself) as a result of grinding, a cone is obtained, the larger base of which is directed towards the headstock; when the rear center is shifted in the opposite direction, i.e., away 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 axis of the headstock spindle 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 conical part of the part in mm.
Example 8. Determine the offset of the center of the tailstock for turning a truncated cone if D = 100 mm, d = 80 mm, L = 300 mm and l = 200 mm. Using formula (12) we find:
The tailstock housing is shifted using divisions 1 (Fig. 206) marked on the end of the base plate, and mark 2 on the end of the tailstock housing.
If there are no divisions at the end of the plate, then move the tailstock body using a measuring ruler, as shown in Fig. 207.
The advantage of machining conical surfaces by displacing the tailstock body is that this method can be used to turn long cones and grind with mechanical feed.
Disadvantages of this method: inability to bore conical holes; loss of time for rearranging the tailstock; the ability to process only shallow cones; misalignment of the centers in the center holes, which leads to rapid and uneven wear of the centers and center holes and causes defects during the secondary installation of the part in the same center holes.
Uneven wear of the center holes can be avoided if a special ball center is used instead of the usual one (Fig. 208). Such centers are mainly used when processing precision cones.
5. Machining conical surfaces using a conical ruler
To process conical surfaces with a slope angle of up to 10-12°, modern lathes usually have a special device called a cone ruler. The scheme for processing a cone using a cone ruler is shown in Fig. 209.
A plate 11 is attached to the machine bed, on which a conical ruler 9 is mounted. The ruler can be rotated around pin 8 at the required angle a to the axis of the workpiece. To secure the ruler in the required position, two bolts 4 and 10 are used. A slider 7 slides freely along the ruler, connecting to the lower transverse part 12 of the caliper using a rod 5 and a clamp 6. So that this part of the caliper can slide freely along the guides, it is disconnected from the carriage 3 by unscrewing the cross screw or disconnecting its nut from the caliper.
If you give 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 attached to the transverse slide of the caliper, they, together with the cutter, will move parallel to the ruler 9. Thanks to this, the cutter will process a conical surface with an inclination angle , equal to the angle α of rotation of the conical ruler.
After each pass, the cutter is set to the cutting depth using the handle 1 of the upper part 2 of the caliper. 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 using formula (11).
Having calculated the value of tan α, it is easy to determine the value of angle α using the table of tangents.
The use of a cone ruler has a number of advantages:
1) setting up the ruler is convenient and quick;
2) when switching to processing cones, there is no need to disrupt the normal setup of the machine, i.e., there is no need to move the tailstock body; the centers of the machine remain in the normal position, i.e. on the same axis, due to which the center holes in the part and the centers of the machine do not work;
3) using a conical ruler, you can not only grind the outer conical surfaces, but also bore conical holes;
4) it is possible to work with a longitudinal self-propelled machine, which increases labor productivity and improves the quality of processing.
The disadvantage of a tapered ruler is the need to disconnect the caliper 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 the gear wheels of the transverse self-propelled machine.
6. Machining conical surfaces with a wide cutter
Machining of conical surfaces (external and internal) with a short cone length can be done with a wide cutter with a plan angle corresponding to the slope angle α of the cone (Fig. 210). The cutter feed can be longitudinal or transverse.
However, the use of a wide cutter on conventional machines is only possible with a cone length not exceeding approximately 20 mm. Wider cutters can only be used 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
Machining tapered holes is one of the most difficult turning jobs; it is much more difficult than processing external cones.
The machining of conical holes on lathes is in most cases carried out by boring with a cutter with turning 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 the tapered ruler are performed in the same way as when turning the outer conical surfaces.
If the hole must be in solid material, then first a cylindrical hole is drilled, which is then bored into a cone with a cutter or machined 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 this, the hole is drilled with one (Fig. 211, b) or two (Fig. 211, c) drills to obtain steps.
After finishing boring the cone, it is reamed using a conical reamer of the appropriate taper. For cones with a small taper, it is more profitable to process the conical holes immediately after drilling with a set of special reamers, as shown in Fig. 212.
8. Cutting modes when processing holes with conical reamers
Conical reamers work under more difficult conditions than cylindrical reamers: while cylindrical reamers remove a slight allowance with small cutting edges, conical reamers cut the entire length of their cutting edges located on the generatrix of the cone. Therefore, when working with conical reamers, feeds and cutting speeds are used less than when working with cylindrical reamers.
When processing holes with conical reamers, the feed is done manually by rotating the tailstock handwheel. It is necessary to ensure that the tailstock quill moves evenly.
Feed when reaming steel is 0.1-0.2 mm/rev, when reaming cast iron 0.2-0.4 mm/rev.
The cutting speed when reaming conical holes with high-speed steel reamers is 6-10 m/min.
Cooling should be used to facilitate the operation of conical reamers and to obtain a clean, smooth surface. When processing steel and cast iron, an emulsion or sulfofresol is used.
9. Measuring conical surfaces
The surfaces of the cones are checked with templates and gauges; measuring and simultaneously checking the angles of the cone is carried out using protractors. In Fig. 213 shows a method for checking a cone using a template.
The outer and inner corners 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 an arc 130. A ruler 5 is rigidly attached to the base 1. Sector 4 moves along the arc of the base, carrying a vernier 3. A square 2 can be attached to the sector 4 by means of a holder 7, in which, in turn, a removable ruler 5 is fixed. The square 2 and the removable ruler 5 have ability to move along the edge of sector 4.
Through 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 on the vernier is 2". The reading obtained when measuring angles is made using the scale and vernier (Fig. 215) as follows: the zero stroke of the vernier shows the number of degrees, and the vernier stroke, coinciding with the stroke of the base scale, shows the number of minutes. In Fig. 215, the 11th stroke of the vernier coincides with the stroke of the base scale, which means 2"X 11 = 22". Therefore, the angle in this case is 76°22".
In Fig. 216 shows combinations of measuring parts of a universal protractor, allowing the measurement of various angles from 0 to 320°.
For more accurate testing of cones in mass production, special gauges are used. In Fig. 217, and shows a conical bushing gauge for checking outer cones, and in Fig. 217, b-conical plug gauge for checking conical holes.
On the gauges, ledges 1 and 2 are made at the ends or marks 3 are applied, which serve to determine the accuracy of the surfaces being checked.
On the. rice. 218 provides an example of checking a conical hole with a plug gauge.
To check the hole, a gauge (see Fig. 218), which has a ledge 1 at a certain distance from the end 2 and two marks 3, is inserted with light pressure into the hole and checked to see if the gauge is swinging in the hole. No wobble indicates that the cone angle is correct. Once you are sure that the angle of the cone is correct, proceed to check its size. To do this, observe to what point the gauge will enter the part being tested. If the end of the part's cone coincides with the left end of ledge 1 or with one of the marks 3 or is between the marks, then the dimensions of the cone are correct. But it may happen that the gauge enters the part so deeply that both marks 3 enter the hole or both ends of the ledge 1 come out of it. This indicates that the hole diameter is larger than specified. If, on the contrary, both risks are outside the hole or none of the ends of the ledge come out of it, then the diameter of the hole is less than the required one.
To accurately check the taper, use the following method. On the surface of the part or gauge to be measured, draw two or three lines with chalk or a pencil along the generatrix of the cone, then insert or put the gauge on the part and turn it part of the turn. If the lines are erased unevenly, this means that the cone of the part is not processed accurately and needs to be corrected. The erasing of lines at the ends of the gauge indicates an incorrect taper; the erasing of the lines in the middle part of the caliber shows that the taper has a slight concavity, which is usually caused by the inaccurate location of the tip of the cutter along 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 the part or gauge. This method gives greater measurement accuracy.
10. Defects in the processing of conical surfaces and measures to prevent them
When processing conical surfaces, in addition to the mentioned types of defects for cylindrical surfaces, the following types of defects are additionally possible:
1) incorrect taper;
2) deviations in the dimensions of the cone;
3) deviations in 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 tailstock housing misalignment, inaccurate rotation of the upper part of the caliper, incorrect installation of the taper ruler, incorrect sharpening or installation of the wide cutter. Therefore, by accurately positioning the tailstock housing, the upper part of the caliper or the cone ruler before starting processing, defects can be prevented. This type of defect can be corrected only if the error along the entire length of the cone is directed into the body of the part, that is, all the diameters of the bushing are smaller, and those of the conical rod are larger than required.
2. The wrong size of the cone with the correct angle, i.e., the wrong size of the diameters along the entire length of the cone, occurs if not enough or too much material is removed. Defects can be prevented only by carefully setting the cutting depth along the dial on finishing passes. We will correct the defect if not enough material was filmed.
3. It may turn out that with the correct taper and exact dimensions of one end of the cone, the diameter of the second end is incorrect. The only reason is failure to comply with the required length of the entire conical section of the part. We will correct the defect if the part is too long. To avoid this type of defect, it is necessary to carefully check its length before processing the cone.
4. Non-straightness of the generatrix of the cone being processed 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 greatly exaggerated form). Thus, this type of defect is the result of the inattentive work of the turner.
Control questions 1. In what ways can conical surfaces be processed on lathes?
2. In what cases is it recommended to rotate the upper part of the caliper?
3. How is the angle of rotation of the upper part of the support for turning a cone calculated?
4. How do you check that the top of the caliper is rotated correctly?
5. How to check the displacement of the tailstock housing?. How to calculate the amount of displacement?
6. What are the main elements of a cone ruler? How to set up a tapered ruler for this part?
7. Set the following angles on the universal protractor: 50°25"; 45°50"; 75°35".
8. What tools are used to measure conical surfaces?
9. Why are there ledges or risks on conical gauges and how to use them?
10. List the types of defects when processing conical surfaces. How to avoid them?
In this lesson we will get acquainted with such a figure as a cone. Let's study the elements of a cone and the types of its sections. And we will find out which figure the cone has many properties in common with.
Fig.1. Cone-shaped objects
In the world, a huge number of things are shaped like a cone. Often we don't even notice them. Road cones warning of road works, roofs of castles and houses, ice cream cones - all these objects are shaped like a cone (see Fig. 1).
Rice. 2. Right Triangle
Consider an arbitrary right triangle with legs and (see Fig. 2).
Rice. 3. Straight circular cone
By rotating a given triangle around one of the legs (without loss of generality, let it be a leg), the hypotenuse will describe the surface, and the leg will describe the circle. Thus, a body will be obtained that is called a right circular cone (see Fig. 3).
Rice. 4. Types of cones
Since we are talking about a straight circular cone, apparently there is both an indirect and a non-circular one? If the base of a cone is a circle, but the vertex is not projected into the center of this circle, then such a cone is called inclined. If the base is not a circle, but an arbitrary figure, then such a body is also sometimes called a cone, but, of course, not circular (see Fig. 4).
Thus, we again come to the analogy already familiar to us from working with cylinders. In fact, a cone is something like a pyramid, it’s just that the pyramid has a polygon at the base, and the cone (which we will consider) has a circle (see Fig. 5).
The segment of the axis of rotation (in our case this is the leg) enclosed inside the cone is called the axis of the cone (see Fig. 6).
Rice. 5. Cone and pyramid
Rice. 6. - cone axis
Rice. 7. Base of the cone
The circle formed by the rotation of the second leg () is called the base of the cone (see Fig. 7).
And the length of this leg is the radius of the base of the cone (or, more simply, the radius of the cone) (see Fig. 8).
Rice. 8. - cone radius
Rice. 9. - top of the cone
The vertex of an acute angle of a rotating triangle lying on the axis of rotation is called the vertex of a cone (see Fig. 9).
Rice. 10. - cone height
The height of a cone is a segment drawn from the top of the cone perpendicular to its base (see Fig. 10).
Here you may have a question: how then does the segment of the axis of rotation differ from the height of the cone? In fact, they coincide only in the case of a straight cone; if you look at an inclined cone, you will notice that these are two completely different segments (see Fig. 11).
Rice. 11. Height in an inclined cone
Let's go back to the straight cone.
Rice. 12. Generators of the cone
The segments connecting the vertex of the cone with the points of the circle of its base are called generators of the cone. By the way, all the generatrices of a right cone are equal to each other (see Fig. 12).
Rice. 13. Natural cone-like objects
Translated from Greek, konos means “pine cone.” In nature there are enough objects that have the shape of a cone: spruce, mountain, anthill, etc. (see Fig. 13).
But we are used to the fact that the cone is straight. It has equal generatrices, and its height coincides with the axis. We called such a cone a straight cone. In school geometry courses, straight cones are usually considered, and by default any cone is considered a right circular one. But we have already said that there are not only straight cones, but also inclined ones.
Rice. 14. Perpendicular section
Let's return to straight cones. “Cut” the cone with a plane perpendicular to the axis (see Fig. 14).
What figure will be on the cut? Of course it's a circle! Let us remember that the plane runs perpendicular to the axis, and therefore parallel to the base, which is a circle.
Rice. 15. Inclined section
Now let's gradually tilt the section plane. Then our circle will begin to gradually turn into an increasingly elongated oval. But only until the section plane collides with the base circle (see Fig. 15).
Rice. 16. Types of sections using the example of a carrot
Those who like to explore the world experimentally can verify this with the help of a carrot and a knife (try cutting slices from a carrot at different angles) (see Fig. 16).
Rice. 17. Axial section of the cone
The section of a cone by a plane passing through its axis is called the axial section of the cone (see Fig. 17).
Rice. 18. Isosceles triangle - sectional figure
Here we get a completely different sectional figure: a triangle. This triangle is isosceles (see Fig. 18).
In this lesson we learned about the cylindrical surface, types of cylinder, elements of a cylinder and the similarity of a cylinder to a prism.
The generatrix of the cone is 12 cm and inclined to the plane of the base at an angle of 30 degrees. Find the axial cross-sectional area of the cone.
Solution
Let us consider the required axial section. This is an isosceles triangle in which the sides are 12 degrees and the base angle is 30 degrees. Then you can proceed in different ways. Or you can draw the height, find it (half of the hypotenuse, 6), then the base (using the Pythagorean theorem), and then the area.
Rice. 19. Illustration for the problem
Or immediately find the angle at the vertex - 120 degrees - and calculate the area as the half-product of the sides and the sine of the angle between them (the answer will be the same).
- Geometry. Textbook for grades 10-11. Atanasyan L.S. and others. 18th ed. - M.: Education, 2009. - 255 p.
- Geometry 11th grade, A.V. Pogorelov, M.: Education, 2002
- Workbook on geometry 11th grade, V.F. Butuzov, Yu.A. Glazkov
- Yaklass.ru ().
- Uztest.ru ().
- Bitclass.ru ().
Homework
Definition. Top of the cone is the point (K) from which the rays originate.
Definition. Cone base is the plane formed by the intersection of a flat surface and all the rays emanating from the top of the cone. A cone can have bases such as circle, ellipse, hyperbola and parabola.
Definition. Generatrix of the cone(L) is any segment that connects the vertex of the cone with the boundary of the base of the cone. The generatrix is a segment of the ray emerging from the vertex of the cone.
Formula. Generator length(L) of a right circular cone through the radius R and height H (via the Pythagorean theorem):
Definition. Guide cone is a curve that describes the contour of the base of the cone.
Definition. Side surface cone is the totality of all the constituents of the cone. That is, the surface that is formed by the movement of the generatrix along the cone guide.
Definition. Surface The cone consists of the side surface and the base of the cone.
Definition. Height cone (H) is a 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 in the diameters of the cross sections D and d of the truncated cone to the distance between them: where R is the radius of the base, and H is the height of the cone.