Formulate the definition of a truncated cone of its elements. Frustum
Conical surface is the surface formed by all straight lines passing through each point of a given curve and a point outside the curve (Fig. 32).
This curve is called guide , straight – forming , dot - top conical surface.
Straight circular conical surface is the surface formed by all straight lines passing through each point of a given circle and a point on a straight line that is perpendicular to the plane of the circle and passes through its center. In what follows we will briefly call this surface conical surface (Fig. 33).
Cone (straight circular cone ) is a geometric body bounded by a conical surface and a plane that is parallel to the plane of the guide circle (Fig. 34).
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Rice. 32 Fig. 33 Fig. 34
A cone can be considered as a body obtained by rotating a right triangle around an axis containing one of the legs of the triangle.
The circle enclosing a cone is called basis . The vertex of a conical surface is called top cone The segment connecting the vertex of a cone with the center of its base is called height cone The segments forming a conical surface are called forming cone Axis of a cone is a straight line passing through the top of the cone and the center of its base. Axial section called the section passing through the axis of the cone. Side surface development A cone is called a sector, the radius of which is equal to the length of the generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone.
The correct formulas for a cone are:
Where R– radius of the base;
H- height;
l– length of the generatrix;
S base– base area;
S side
S full
V– volume of the cone.
Truncated cone called the part of the cone enclosed between the base and the cutting plane parallel to the base of the cone (Fig. 35).
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A truncated cone can be considered as a body obtained by rotating a rectangular trapezoid around an axis containing the side of the trapezoid perpendicular to the bases.
The two circles enclosing a cone are called its reasons . Height of a truncated cone is the distance between its bases. The segments forming the conical surface of a truncated cone are called forming . A straight line passing through the centers of the bases is called axis truncated cone. Axial section called the section passing through the axis of a truncated cone.
For a truncated cone the correct formulas are:
(8)
Where R– radius of the lower base;
r– radius of the upper base;
H– height, l – length of the generatrix;
S side– lateral surface area;
S full– total surface area;
V– volume of a truncated cone.
Example 1. The cross section of the cone parallel to the base divides the height in a ratio of 1:3, counting from the top. Find the lateral surface area of a truncated cone if the radius of the base and the height of the cone are 9 cm and 12 cm.
Solution. Let's make a drawing (Fig. 36).
To calculate the area of the lateral surface of a truncated cone, we use formula (8). Let's find the radii of the bases About 1 A And About 1 V and forming AB.
Consider similar triangles SO2B And SO 1 A, similarity coefficient, then 
From here 
Since then 
The lateral surface area of a truncated cone is equal to:
Answer: .
Example 2. A quarter circle of radius is folded into a conical surface. Find the radius of the base and the height of the cone.
Solution. The quadrant of the circle is the development of the lateral surface of the cone. Let's denote r– radius of its base, H – height. Let's calculate the lateral surface area using the formula: . It is equal to the area of a quarter circle: . We get an equation with two unknowns r And l(forming a cone). In this case, the generatrix is equal to the radius of the quarter circle R, which means we get the following equation: , from where Knowing the radius of the base and the generator, we find the height of the cone:
Answer: 2 cm, .
Example 3. A rectangular trapezoid with an acute angle of 45 O, a smaller base of 3 cm and an inclined side equal to , rotates around the side perpendicular to the bases. Find the volume of the resulting body of revolution.
Solution. Let's make a drawing (Fig. 37).
As a result of rotation, we obtain a truncated cone; to find its volume, we calculate the radius of the larger base and height. In the trapeze O 1 O 2 AB we will conduct AC^O 1 B. B we have: this means that this triangle is isosceles A.C.=B.C.=3 cm.
Answer:
Example 4. A triangle with sides 13 cm, 37 cm and 40 cm rotates around an external axis, which is parallel to the larger side and located at a distance of 3 cm from it (the axis is located in the plane of the triangle). Find the surface area of the resulting body of revolution.
Solution
.
Let's make a drawing (Fig. 38).
The surface of the resulting body of revolution consists of the lateral surfaces of two truncated cones and the lateral surface of a cylinder. In order to calculate these areas, it is necessary to know the radii of the bases of the cones and the cylinder ( BE And O.C.), forming cones ( B.C. And A.C.) and cylinder height ( AB). The only unknown is CO. 
this is the distance from the side of the triangle to the axis of rotation. We'll find DC. The area of triangle ABC on one side is equal to the product of half the side AB and the altitude drawn to it DC, on the other hand, knowing all the sides of the triangle, we calculate its area using Heron’s formula.
Introduction
Rice. 1. Objects from life that have the shape of a truncated ko-nu-sa
Where do you think new figures come from in geometry? It’s all very simple: a person in life has become with similar objects and comes, as if call them. Let's look at the cabinet on which the lions in the circus sit, a piece of carrots that come out when we're just... -a part of it, an active volcano and, for example, light from the fo-na-ri-ka (see Fig. 1).
Truncated cone, its elements and axial section
Rice. 2. Geo-met-ri-che-fi-gu-ry
We see that all these figures are of a similar shape - both from below and from above they are bounded by circles, but they narrow towards the top ( see fig. 2).
Rice. 3. From the upper part of the co-nu-sa
It looks like a cone. Just not enough top-hush. We mentally imagine that we take a cone and remove the upper part from it with one swing of a sharp sword (see Fig. 3).
Rice. 4. Truncated cone
This is exactly our figure; it is called a truncated cone (see Fig. 4).
Rice. 5. Se-che-nie, parallel-os-no-va-niyu ko-nu-sa
Let a cone be given. Let's create a plane, a parallel plane of the axis of this co-nu-sa and a cross-cutting cone (see. Fig. 5).
It will split the cone into two bodies: one of them is a cone of smaller size, and the second is called a truncated cone ( see Fig. 6).
Rice. 6. Obtained bodies in a parallel section
Thus, a truncated cone is a part of the cone, connected between its main body and the parallel main body. but flat. As in the case of a cone, a truncated cone can have a circle as its base - in this case it is called a circle. If the original cone was straight, then the truncated cone is called straight. As in the case of ko-nu-sa-mi, we will look at the keys, but straight circular truncated ko-nu-s sy, if it is not specifically indicated that we are talking about an indirect truncated co-nu-se or in its basis there are no circles.
Rice. 7. Rotation of a rectangular trap
Our global theme is bodies of rotation. A truncated cone is not an exception! Let us remember that in order to obtain a co-nu-sa, we smo-mat-ri-va-li a rectangular triangle and rotate it around ka-te-ta? If the resulting cone is cut with a plane parallel to the axis, then there will be no straight line left from the triangle -mo-coal trap. Its rotation around the smaller side will give us a truncated cone. Let us note again that we are, obviously, talking only about a direct circular co-nu-se (see Fig. 7).
Rice. 8. Os-no-va-niya truncated-no-go ko-nu-sa
I’ll make a few preparations. The basis of the half-ko-nu-sa and the circle, half-cha-yu-shay in the section of the ko-nu-sa flat, on- they call os-no-va-ni-ya-mi truncated ko-nu-sa (lower and upper) (see Fig. 8).
Rice. 9. Ob-ra-zu-yu-schi truncated ko-nu-sa
From the cuttings of the ra-zu-yu-shih half of the co-nu-sa, connected between the os-but-va-ni-mi truncated-but- go ko-nu-sa, they call about-ra-zu-yu-schi-mi truncated-no-go ko-nu-sa. Since all educational outcomes are equal and all educational outcomes are from the same are equal, then the ob-ra-zu-yu truncated co-nu-sa are equal (do not confuse the truncated and the truncated!). From here follows the equality of the tra-pe-tion of the axis of the sec-tion (see Fig. 9).
From the axis of rotation, enclosed inside the truncated co-nu-sa, they call it the axis of the truncated axis ko-nu-sa. This re-cut, ra-zu-me-et-sya, unites the centers of its fundamentals (see Fig. 10).
Rice. 10. Axis of truncated ko-nu-sa
You-so-ta truncated ko-nu-sa is a per-pen-di-ku-lyar, pro-ve-den from the point of one of the os-no-va- niya to another base. Most often, in the quality of you, you have truncated its axis.
Rice. 11. Ose-voe se-che-nie truncated-no-go-ko-nu-sa
The axis section of a truncated ko-nu-sa is the section passing through its axis. It has the form of a trapezoid, a little later we will show its equality (see Fig. 11).
Areas of the lateral and total surfaces of a truncated cone
Rice. 12. Cone with introduced symbols
Let's find the area of the bo-co-voy on the top of the truncated ko-nu-sa. Let the bases of the truncated co-nu-sa have radii and , and let the ob-ra-zu-yu be equal (see Fig. 12).
Rice. 13. Designation of the ob-ra-zu-yu-shchei from-se-chen-no-th ko-nu-sa
Let's find the area of the bo-ko-voy on top of the truncated co-nu-sa as the difference in the area of bo-ko-voys on the top-but- ste-khod-no-go ko-nu-sa and from-se-chen-no-go. To do this, we denote through the ob-ra-zu-yu from-se-chen-no-th ko-nu-sa (see Fig. 13).
Then is-ko-may.
Rice. 14. Similar triangles
All that's left is for you to figure it out.
Let us note that from po-do-biy tri-corn-ni-kov, from-to-yes (see Fig. 14).
It would be possible to express this by dividing it into the difference between the radii, but we do not need this, because in the present case it is precisely the fi- gu-ri-ru-et pro-iz-ve-de-nie. Substituting instead of it, we finally have: .
Now it’s not difficult to get a shape for a full surface area. To do this, add exactly the area of the two circles of the bases: .
Task
Rice. 15. Illu-stration to for-da-che
Let the truncated cone be rotated by a rectangular trapezoid around its height. The middle line of the trapezoid is equal to , and the larger side is equal to (see Fig. 15). Find the area of the bo-co-voy on the top-no-sti of the truncated ko-nu-sa.
Solution
From the formula we know that 
.
The formation of the ko-nu-sa will be a large hundred-ro-on-going tra-pe-tion, that is, Ra-di-u-sy ko- well-sa - this is the basis of the tra-pe-tion. We can't find them. But we don’t need it: we only need their sum, and the sum of the bases of a trapezoid is twice as large as its midline, that is, it is equal to . Then .
Similarities between truncated cones and pyramids
Pay attention to the fact that when we talk about co-nu-se, we talk about it between him and pi -ra-mi-doy - the formulas were analogous. It’s the same here, because a truncated cone is very similar to a truncated pi-ra-mi-du, so the formulas for the area are large and complete top-not-stey truncated ko-nu-sa and pi-ra-mi-dy (and soon there will be formulas for volume) analog-lo-gic- us.
Task
Rice. 1. Illu-strat-tion to za-da-che
The ra-di-u-sy os-no-va-niy use-chen-no-go ko-nu-sa are equal to and , and the ob-ra-zu-yu-shchaya is equal to . Find the truncated co-nu-sa and the area of its axis (see Fig. 1).
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 .
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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.
Geometry is a branch of mathematics that studies structures in space and the relationships between them. In turn, it also consists of sections, and one of them is stereometry. It involves the study of the properties of three-dimensional figures located in space: cube, pyramid, ball, cone, cylinder, etc.
A cone is a body in Euclidean space that is bounded by a conical surface and the plane on which the ends of its generators lie. Its formation occurs during the rotation of a right triangle around any of its legs, so it belongs to bodies of revolution.
Components of a cone
There are the following types of cones: oblique (or inclined) and straight. Oblique is one whose axis does not intersect with the center of its base at a right angle. For this reason, the height in such a cone does not coincide with the axis, since it is a segment that is lowered from the top of the body to the plane of its base at an angle of 90°.
The cone whose axis is perpendicular to its base is called straight. The axis and height in such a geometric body coincide due to the fact that the vertex in it is located above the center of the diameter of the base.
The cone consists of the following elements:
- The circle that is its base.
- Lateral surface.
- A point not lying in the plane of the base, called the vertex of the cone.
- Segments that connect the points of the circle of the base of a geometric body and its vertex.
All these segments are generators of the cone. They are inclined to the base of the geometric body, and in the case of a right cone, their projections are equal, since the vertex is equidistant from the points of the circle of the base. Thus, we can conclude that in a regular (straight) cone the generators are equal, that is, they have the same length and form the same angles with the axis (or height) and the base.
Since in an oblique (or inclined) body of rotation the vertex is shifted relative to the center of the base plane, the generators in such a body have different lengths and projections, since each of them is at a different distance from any two points of the circle of the base. In addition, the angles between them and the height of the cone will also be different.
Length of generatrices in a straight cone
As written earlier, the height in a right geometric body of revolution is perpendicular to the plane of the base. Thus, the generatrix, height and radius of the base create a right triangle in the cone.
That is, knowing the base radius and height, using the formula from the Pythagorean theorem, you can calculate the length of the generatrix, which will be equal to the sum of the squares of the base radius and height:
l 2 = r 2 + h 2 or l = √r 2 + h 2
where l is the generator;
r - radius;
h - height.
Generator in an inclined cone
Based on the fact that in an oblique or inclined cone the generators do not have the same length, it will not be possible to calculate them without additional constructions and calculations.
First of all, you need to know the height, axis length and base radius.
r 1 = √k 2 - h 2
where r 1 is the part of the radius between the axis and the height;
k - axis length;
h - height.
As a result of adding the radius (r) and its part lying between the axis and height (r 1), you can find out the complete generated generatrix of the cone, its height and part of the diameter:
where R is the leg of a triangle formed by the height, the generator and part of the diameter of the base;
r - radius of the base;
r 1 - part of the radius between the axis and the height.
Using the same formula from the Pythagorean theorem, you can find the length of the generatrix of the cone:
l = √h 2 + R 2
or, without separately calculating R, combine the two formulas into one:
l = √h 2 + (r + r 1) 2.
Regardless of whether the cone is straight or oblique and what the input data are, all methods for finding the length of the generatrix always come down to one result - the use of the Pythagorean theorem.
Cone section
Axial is a plane passing along its axis or height. In a straight cone, such a section is an isosceles triangle, in which the height of the triangle is the height of the body, its sides are the generators, and the base is the diameter of the base. In an equilateral geometric body, the axial section is an equilateral triangle, since in this cone the diameter of the base and the generators are equal.
The plane of the axial section in a straight cone is the plane of its symmetry. The reason for this is that its top is located above the center of its base, that is, the plane of the axial section divides the cone into two identical parts.
Since the height and axis do not coincide in an inclined volumetric body, the axial section plane may not include the height. If many axial sections in such a cone can be constructed, since for this only one condition must be met - it must pass only through the axis, then the axial section of the plane to which the height of this cone will belong can be drawn only one, because the number of conditions increases, and, as is known, two straight lines (together) can belong to only one plane.
Cross-sectional area
The previously mentioned axial section of the cone is a triangle. Based on this, its area can be calculated using the formula for the area of a triangle:
S = 1/2 * d * h or S = 1/2 * 2r * h
where S is the cross-sectional area;
d - base diameter;
r - radius;
h - height.
In an oblique or inclined cone, the cross-section along the axis is also a triangle, so the cross-sectional area in it is calculated in a similar way.
Volume
Since a cone is a three-dimensional figure in three-dimensional space, its volume can be calculated. The volume of a cone is a number that characterizes this body in a unit of volume, that is, in m3. The calculation does not depend on whether it is straight or oblique (oblique), since the formulas for these two types of bodies do not differ.
As stated earlier, the formation of a right cone occurs due to the rotation of a right triangle along one of its legs. An inclined or oblique cone is formed differently, since its height is shifted away from the center of the plane of the base of the body. Nevertheless, such differences in structure do not affect the method for calculating its volume.
Volume calculation
Any cone looks like this:
V = 1/3 * π * h * r 2
where V is the volume of the cone;
h - height;
r - radius;
π is a constant equal to 3.14.
To calculate the height of a body, you need to know the radius of the base and the length of its generatrix. Since the radius, height and generator are combined into a right triangle, the height can be calculated using the formula from the Pythagorean theorem (a 2 + b 2 = c 2 or in our case h 2 + r 2 = l 2, where l is the generator). The height will be calculated by taking the square root of the difference between the squares of the hypotenuse and the other leg:
a = √c 2 - b 2
That is, the height of the cone will be equal to the value obtained after taking the square root of the difference between the square of the length of the generatrix and the square of the radius of the base:
h = √l 2 - r 2
By calculating the height using this method and knowing the radius of its base, you can calculate the volume of the cone. The generator plays an important role in this case, since it serves as an auxiliary element in the calculations.
Similarly, if the height of a body and the length of its generatrice are known, one can find out the radius of its base by taking the square root of the difference between the square of the generatrix and the square of the height:
r = √l 2 - h 2
Then, using the same formula as above, calculate the volume of the cone.
Volume of an inclined cone
Since the formula for the volume of a cone is the same for all types of bodies of rotation, the difference in its calculation is the search for height.
In order to find out the height of an inclined cone, the input data must include the length of the generatrix, the radius of the base, and the distance between the center of the base and the intersection of the height of the body with the plane of its base. Knowing this, you can easily calculate that part of the base diameter that will be the base of a right triangle (formed by the height, the generatrix and the plane of the base). Then, again using the Pythagorean theorem, calculate the height of the cone, and subsequently its volume.
Rice. 1. Objects from life that have the shape of a truncated cone
Where do you think new shapes come from in geometry? Everything is very simple: a person comes across similar objects in life and comes up with a name for them. Let's consider a stand on which lions sit in a circus, a piece of carrot that is obtained when we cut only part of it, an active volcano and, for example, the light from a flashlight (see Fig. 1).
Rice. 2. Geometric shapes
We see that all these figures are of a similar shape - both below and above they are limited by circles, but they taper upward (see Fig. 2).
Rice. 3. Cutting off the top of the cone
It looks like a cone. The top is just missing. Let's mentally imagine that we take a cone and cut off the upper part from it with one swing of a sharp sword (see Fig. 3).
Rice. 4. Truncated cone
The result is exactly our figure, it is called a truncated cone (see Fig. 4).
Rice. 5. Section parallel to the base of the cone
Let a cone be given. Let us draw a plane parallel to the plane of the base of this cone and intersecting the cone (see Fig. 5).
It will split the cone into two bodies: one of them is a smaller cone, and the second is called a truncated cone (see Fig. 6).
Rice. 6. The resulting bodies with a parallel section
Thus, a truncated cone is a part of a cone enclosed between its base and a plane parallel to the base. As with a cone, a truncated cone can have a circle at its base, in which case it is called circular. If the original cone was straight, then the truncated cone is called straight. As in the case of cones, we will consider exclusively straight circular truncated cones, unless it is specifically stated that we are talking about an indirect truncated cone or its bases are not circles.
Rice. 7. Rotation of a rectangular trapezoid
Our global topic is bodies of rotation. The truncated cone is no exception! Let us remember that to obtain a cone we considered a right triangle and rotated it around the leg? If the resulting cone is intersected by a plane parallel to the base, then the triangle will remain a rectangular trapezoid. Its rotation around the smaller side will give us a truncated cone. Let us note again that, of course, we are talking only about a straight circular cone (see Fig. 7).
Rice. 8. Bases of a truncated cone
Let's make a few comments. The base of a complete cone and the circle resulting from a section of the cone by a plane are called the bases of a truncated cone (lower and upper) (see Fig. 8).
Rice. 9. Generators of a truncated cone
The segments of the generators of a complete cone, enclosed between the bases of a truncated cone, are called generators of a truncated cone. Since all the generators of the original cone are equal and all the generators of the cut off cone are equal, then the generators of the truncated cone are equal (do not confuse the cut off and truncated one!). This implies that the axial section of the trapezoid is isosceles (see Fig. 9).
The segment of the axis of rotation enclosed inside a truncated cone is called the axis of the truncated cone. This segment, of course, connects the centers of its bases (see Fig. 10).
Rice. 10. Axis of a truncated cone
The height of a truncated cone is a perpendicular drawn from a point of one of the bases to the other base. Most often, the height of a truncated cone is considered to be its axis.
Rice. 11. Axial section of a truncated cone
The axial section of a truncated cone is the section passing through its axis. It has the shape of a trapezoid; a little later we will prove that it is isosceles (see Fig. 11).
Rice. 12. Cone with introduced notations
Let us find the area of the lateral surface of the truncated cone. Let the bases of the truncated cone have radii and , and the generatrix be equal (see Fig. 12).
Rice. 13. Designation of the generatrix of the cut off cone
Let us find the area of the lateral surface of the truncated cone as the difference between the areas of the lateral surfaces of the original cone and the cut off one. To do this, let us denote by the generatrix of the cut off cone (see Fig. 13).
Then what you are looking for.
Rice. 14. Similar triangles
All that remains is to express.
Note that from the similarity of triangles, whence (see Fig. 14).
It would be possible to express , dividing by the difference of the radii, but we do not need this, because the product we are looking for appears in the desired expression. Substituting , we finally have: .
It is now easy to obtain a formula for the total surface area. To do this, just add the area of the two circles of the bases: .
Rice. 15. Illustration for the problem
Let a truncated cone be obtained by rotating a rectangular trapezoid around its height. The middle line of the trapezoid is equal, and the large side is (see Fig. 15). Find the lateral surface area of the resulting truncated cone.
Solution
From the formula we know that 
.
The generatrix of the cone will be the larger side of the original trapezoid, that is, the radii of the cone are the bases of the trapezoid. We can't find them. But we don’t need it: we only need their sum, and the sum of the bases of a trapezoid is twice as large as its midline, that is, it is equal to . Then .
Please note that when we talked about the cone, we drew parallels between it and the pyramid - the formulas were similar. It’s the same here, because a truncated cone is very similar to a truncated pyramid, so the formulas for the areas of the lateral and total surfaces of a truncated cone and pyramid (and soon there will be formulas for volume) are similar.
Rice. 1. Illustration for the problem
The radii of the bases of the truncated cone are equal to and , and the generatrix is equal to . Find the height of the truncated cone and the area of its axial section (see Fig. 1).