Consider any line l (curve or broken line) lying in a certain plane (Fig. 386, a, b), and an arbitrary point M not lying in this plane. All possible straight lines connecting the point M with all points of the line form a surface a; such a surface is called a conical surface, a point is a vertex, a line is called a guide, straight lines are generators. On fig. 386 we do not limit the surface to its top, but imagine it extending indefinitely on both sides of the top.
If the conical surface is cut by some plane parallel to the plane of the guide, then in the section we get a line (curve or broken line, depending on whether it was a curve or a broken line), homothetic to the line l, with the homothety center at the top of the conical surface. Indeed, the ratio of any corresponding line segments will be constant:
So, sections of a conical surface by planes parallel to the plane of the guide are similar and similarly located, with the center of similarity at the top of the conical surface; the same is true for any parallel planes that do not pass through a surface vertex.
Now let the guide be a closed convex line (curve in Fig. 387, a, broken line in Fig. 387, b). A body bounded laterally by a conical 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 (if it is a curved line) or a pyramid (if it is a broken line).
Pyramids are classified according to the number of sides of the polygon that lies at their base. They talk about triangular, quadrangular and generally -angular pyramids. Note that the -coal pyramid has a face: side faces and a base. At the top of the pyramid, we have a -hedral angle with flat and dihedral angles.
They are respectively called flat vertex angles and dihedral angles at lateral edges. At the tops of the base we have trihedral angles; their flat angles formed by the sides, edges and sides of the base are called flat angles at the base, dihedral angles between the side faces and the plane of the base are called dihedral angles at the base.
A triangular pyramid is otherwise called a tetrahedron (that is, a tetrahedron). Any of its faces can be taken as a base.
A pyramid is called regular if two conditions are met: 1) a regular polygon lies at the base of the pyramid,
2) the height lowered from the top of the pyramid to the base intersects it at the center of this polygon (in other words, the top of the pyramid is projected into the center of the base).
Note that a regular pyramid is not, generally speaking, a regular polyhedron!
We note some properties of a regular -coal pyramid. Let us draw the height SO through the top of such a pyramid (Fig. 388).
Let's rotate the entire pyramid as a whole around this height by an angle. With such a rotation, the base polygon will turn into itself: each of its vertices will take the position of the neighboring one. The top of the pyramid and its height (axis of rotation!) will remain in place, and therefore the pyramid as a whole will be combined with itself: each side edge will go to the next one, each side face will be combined with the next one, each dihedral angle at the side edge will also be combined with the next one.
Hence the conclusion: all side edges are equal to each other, all side faces are equal isosceles triangles, all dihedral angles at the base are equal, all flat angles at the top are equal, all flat angles at the base are equal.
From the number of cones in the course of elementary geometry, we study a right circular cone, that is, a cone whose base is a circle, and whose vertex is projected into the center of this circle.
A straight circular cone is shown in fig. 389. If we draw height SO through the vertex of a cone and rotate the cone about this height by an arbitrary angle, then the circumference of the base will slide by itself; the height and vertex will remain in place, so when rotated to any angle, the cone will align with itself. From this it can be seen, in particular, that all the generators of the cone are equal to each other and are equally inclined to the plane of the base. Sections of the cone by planes passing through its height will be isosceles triangles equal to each other. The entire cone is obtained by rotating the right triangle SOA around its leg (which becomes the height of the cone). Therefore, a right circular cone is a body of revolution and is also called a cone of revolution. Unless otherwise stated, for brevity we will hereafter simply say "cone", meaning by this a cone of revolution.
The sections of a cone by planes parallel to the plane of its base are circles (if only because they are homothetic to the circle of the base).
A task. The dihedral angles at the base of a regular triangular pyramid are a. Find the dihedral angles at the side edges.
Solution. Let us temporarily designate the side of the base of the pyramid as a. Let's draw a section of the pyramid by a plane containing its height SO and the median of the base AM (Fig. 390).
A cone (more precisely, a circular cone) is a body that consists 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 connecting the top of the cone with the points of the base (Fig. 1) Segments connecting the top of the cone with points of the circumference of the base, are called generatrices, of the cone. All generators of the cone are equal to each other. The surface of the cone consists of a base and a side surface.
Rice. one
A cone is called straight if the line connecting the vertex of the cone with the center of the base is perpendicular to the plane of the base. Visually, a straight circular cone can be imagined as a body obtained by rotating a right triangle around its leg as an axis (Fig. 2).
Rice. 2
The height of a cone is the perpendicular drawn from its top to the plane of its base. For a right cone, the base of the height coincides with the center of the base. The axis of a right circular cone is called a straight line containing its height.
The section of a cone by a plane passing through its apex is an isosceles triangle, in which the sides are generators of the cone (Fig. 3). In particular, an isosceles triangle is an axial section of a cone. This is a section that passes through the axis of the cone (Fig. 4).
Rice. 3 Fig. four
Cone surface area
The lateral surface of the cone, as well as the lateral surface of the cylinder, can be turned into a plane by cutting it along one of the generators (Fig. 2, a, b). The development of the lateral surface of the cone is a circular sector (Fig. 2.6), the radius of which is equal to the generatrix of the cone, and the length of the arc of the sector is the circumference of the base of the cone.
The area of its development is taken as the area of the lateral surface of the cone. We express the area Sside of the lateral surface of the cone through its generatrix l and the radius of the base r.
The area of the circular sector - the development of the lateral surface of the cone (Fig. 2) - is equal to (Pl2a) / 360, where a is the degree measure of the arc ABA", therefore
Sbok \u003d (Pl2a) / 360. (*)
Let us express a in terms of l and r. Since the length of the arc ABA "is equal to 2Pr (the circumference of the base of the cone), then 2Pr \u003d Pla / 180, from which a \u003d 360r / l. Substituting this expression into the formula (*), we get:
Sside = Prl. (**)
Thus, the area of the lateral surface of the cone is equal to the product of half the circumference of the base and the generatrix.
The total surface area of a cone is the sum of the areas of the lateral surface and the base. To calculate the area Scon of the full surface of the cone, the formula is obtained: Scon \u003d Pr (l + r). (***)
Frustum
Take an arbitrary cone and draw a cutting plane perpendicular to its axis. This plane intersects with the cone in a circle and splits the cone into two parts. One of the parts is a cone, and the other is called a truncated cone. The base of the original cone and the circle obtained in the section of this cone by a plane are called the bases of the truncated cone, and the segment connecting their centers is called the height of the truncated cone.
The part of the conical surface that bounds the truncated cone is called its lateral surface, and the segments of the generatrix of the conical surface enclosed between the bases are called the generators of the truncated cone. All generators of a truncated cone are equal to each other (prove it yourself).
The area of the lateral surface of a truncated cone is equal to the product of half the sum of the circumferences of the bases and the generatrix: Sside \u003d P (r + r1) l.
Additional information about the cone
1. In geology, there is the concept of "removal cone". This is a relief form formed by an accumulation of clastic rocks (pebbles, gravel, sand) carried by mountain rivers to a foothill plain or to a flatter wide valley.
2. In biology, there is the concept of "cone of growth". This is the top of the shoot and root of plants, consisting of cells of the educational tissue.
3. "Cones" is a family of marine molluscs of the subclass of the anterior gills. The shell is conical (2–16 cm), brightly colored. There are over 500 types of cones. They live in the tropics and subtropics, are predators, have a poisonous gland. The bite of the cones is very painful. Known deaths. The shells are used as decorations and souvenirs.
4. According to statistics, 6 people per 1 million inhabitants die annually from lightning discharges on Earth (more often in southern countries). This would not happen if there were lightning rods everywhere, as a safety cone is formed. The higher the lightning rod, the larger the volume of such a cone. Some people try to hide from discharges under a tree, but the tree is not a conductor, charges accumulate on it and the tree can be a source of voltage.
5. In physics, there is the concept of "solid angle". This is a tapered corner carved into the ball. The unit of solid angle is 1 steradian. 1 steradian is a solid angle whose square of radius is equal to the area of the part of the sphere it cuts out. If a light source of 1 candela (1 candle) is placed in this corner, then we get a luminous flux of 1 lumen. The light from a movie camera, a searchlight, spreads in the form of a cone.
Cone (from the Greek "konos")- Pine cone. The cone has been familiar to people since ancient times. In 1906, the book "On the Method", written by Archimedes (287-212 BC), was discovered, in this book a solution is given to the problem of the volume of the common part of intersecting cylinders. Archimedes says that this discovery belongs to the ancient Greek philosopher Democritus (470-380 BC), who, using this principle, obtained formulas for calculating the volume of a pyramid and a cone.
Cone (circular cone) - a body that consists of a circle - the base of the cone, a point that does not belong to the plane of this circle - the top of the cone and all segments connecting the top of the cone and the base circle points. The segments that connect the top of the cone with the points of the circle of the base are called the generators of the cone. The surface of the cone consists of a base and a side surface.
A cone is called straight if the line that connects the vertex of the cone with the center of the base is perpendicular to the plane of the base. A right circular cone can be considered as a body obtained by rotating a right triangle around its leg as an axis.
The height of a cone is the perpendicular drawn from its top to the plane of its base. For a right cone, the base of the height coincides with the center of the base. The axis of a right cone is a straight line containing its height.
The section of a cone by a plane passing through the generatrix of the cone and perpendicular to the axial section drawn through this generatrix is called the tangent plane of the cone.
A plane perpendicular to the axis of the cone intersects the cone in a circle, and the lateral surface in a circle centered on the axis of the cone.
A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The rest is called a truncated cone.
The volume of a cone is equal to one third of the product of the height and the area of the base. Thus, all cones resting on a given base and having a vertex located on a given plane parallel to the base have the same volume, since their heights are equal.
The lateral surface area of a cone can be found using the formula:
S side \u003d πRl,
The total surface area of the cone is found by the formula:
S con \u003d πRl + πR 2,
where R is the radius of the base, l is the length of the generatrix.
The volume of a circular cone is
V = 1/3 πR 2 H,
where R is the radius of the base, H is the height of the cone
The area of the lateral surface of a truncated cone can be found by the formula:
S side = π(R + r)l,
The total surface area of a truncated cone can be found using the formula:
S con \u003d πR 2 + πr 2 + π(R + r)l,
where R is the radius of the lower base, r is the radius of the upper base, l is the length of the generatrix.
The volume of a truncated cone can be found as follows:
V = 1/3 πH(R 2 + Rr + r 2),
where R is the radius of the lower base, r is the radius of the upper base, H is the height of the cone.
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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 a cone
Theorem 1.1. Axial section of the cone
Volume and area :
Theorem 2. Volume of a cone
Theorem 3. The area of the lateral surface of the cone
Frustum :
Theorem 4. Section parallel to the base
Definition 6. Truncated cone
Theorem 5. Volume of a truncated cone
Theorem 6. Area of the lateral surface of a truncated cone
Definition
A body bounded from 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 that does not lie in the plane of the base (top) and all segments connecting the top with the points of the base. 
A right cone is a cone whose height contains the center of the base of the cone as its base.
Consider any line (curve, broken or mixed) (for example, l) lying in some plane, and an arbitrary point (for example, M) not lying in this plane. All possible lines connecting the point M with all points of the given line l, form surface called canonical. The point M is the vertex of such a surface, and the given line l - guide. All lines connecting point M with all points of the line l, called generating. A canonical surface is not limited by its vertex or guide. It extends indefinitely on both sides of the summit. Now let the guide be a closed convex line. If the guide is a broken line, then the body bounded laterally 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 curve or a mixed line, then the body bounded laterally 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 (curve 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 lines passing through the vertex of the cone and any of the points of the curve that bounds the figure of the base of the cone are called generators of the cone. Most often, in geometric problems, a generatrix of a straight line means a segment of this straight line enclosed between the top and the plane of the base of the cone.
The bottom of a limited mixed line is a very rare case. It is listed here only because it can be considered in geometry. The case with a curved guide is more often considered. Although, that the case with an arbitrary curve, that the case with a mixed guide, is of little use and it is difficult to derive any regularities in them. Of the number of cones in the course of elementary geometry, a right circular cone is studied.
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 a circle as a guide, you can define a circular cone.
Definition 2
. A circular cone is a body that consists of a circle (base), a point that does not lie in the plane of the base (top) and all segments connecting the top with the points of the base.
Definition 3
. The height of the cone is the perpendicular dropped from the top to the plane of the base of the cone. It is possible to single out a cone, the height of which falls into the center of the flat figure of the base.
Definition 4
. A right cone is a cone whose height contains the center of the base of the cone as its base.
If we connect these two definitions, we get a cone, the base of which is a circle, and the height falls into the center of this circle.
Definition 5
. A right circular cone is called a cone, the base of which 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 the 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 a 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 lines of the base by definition, perpendicular to the line to the plane. Therefore, the triangles MOA, MOV and MOS are rectangular and are equal in two legs (MO - general, OA \u003d OB \u003d OS - base radii. Therefore, the hypotenuses, i.e. generators, are also equal.
The radius of the base of a cone is sometimes called cone radius. The height of a cone is also called cone axis, so any section passing through a height is called axial section. Any axial section intersects the base in diameter (because 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 the 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.
Today we will tell you about 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 right cone is a figure that results from the rotation of a right 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 a cone is the segment that connects the top of the cone to the center of the base. The generatrix of a cone is a segment that connects the vertex of the cone to any point on the line of the circumference of the base.
Since the cone is formed by the rotation of a right 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 generatrix. And now more about how to find the length of the generatrix of the cone.
Finding a generatrix
The easiest way to understand how to find a generatrix is to use 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 generatrix.
So, the height of the cone (9 cm) is one of the legs of the right triangle, with the help of 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 diameter given to us in half and get the length of the radius: 18:2 = 9. The radius is 9.
Now it is very easy to find the generatrix of the cone. Since it is the 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 generator = 64 (the square of the length of the radius) + 64 (the square of the length of the height) = 64x2 = 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 this. For 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.
