Cone. Frustum

Tapered surface called the surface formed by all straight lines passing through each point of the given curve and a point outside the curve (Fig. 32).

This curve is called guide , direct - generating , dot - summit conical surface.

Straight circular tapered surface called the surface formed by all lines passing through each point of the given circle and a point on the line that is perpendicular to the plane of the circle and passes through its center. In what follows, this surface will be briefly referred to as conical surface (fig.33).

cone (straight circular cone ) is called a geometric body bounded by a conical surface and a plane that is parallel to the plane of the guide circle (Fig. 34).

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 that bounds the cone is called basis . The vertex of a conical surface is called summit cone. The line segment connecting the top of a cone with the center of its base is called height cone. The segments that form a conical surface are called generating cone. axis of a cone is a straight line passing through the vertex of the cone and the center of its base. Axial section called the section passing through the axis of the cone. Lateral surface development cone is called a sector whose radius equal to length generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone.

For a cone, the following formulas are true:

where R is the radius of the base;

H- height;

l- the length of the generatrix;

S main- base area;

S side

S full

V is the 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).

A truncated cone can be considered as a body obtained by rotation rectangular trapezoid about an axis containing the side of the trapezoid perpendicular to the bases.

The two circles that bound the cone are called its grounds . Height of a truncated cone is the distance between its bases. The segments that form the conical surface of a truncated cone are called generating . The straight line passing through the centers of the bases is called axis truncated cone. Axial section called the section passing through the axis of the truncated cone.

For a truncated cone, the following formulas are true:

(8)

where R is the radius of the lower base;

r is the radius of the upper base;

H is the height, l is the length of the generatrix;

S side is the lateral surface area;

S full is the total surface area;

V is the volume of the truncated cone.

Example 1 The section of the cone parallel to the base divides the height in a ratio of 1:3, counting from the top. Find the area of ​​the lateral surface 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). Find the radii of the bases About 1 A and About 1 V and generating AB.

Consider similar triangles SO 2 B and SO 1A, coefficient of similarity , then

From here

Since then

The area of ​​the lateral surface of a truncated cone is equal to:

Answer: .

Example2. 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 quadruple of a circle is a development of the lateral surface of the cone. Denote r is the radius of its base, H- height. The lateral surface area is calculated by the formula: . It is equal to the area of ​​a quarter of a circle: . We get an equation with two unknowns r and l(generator of a cone). In this case, the generatrix is ​​equal to the radius of a quarter of a circle R, so we get the following equation: , whence Knowing the radius of the base and generatrix, we find the height of the cone:

Answer: 2 cm, .

Example 3 Rectangular trapezoid with acute angle 45 O, with a smaller base 3 cm and an inclined side equal to , rotates around the side perpendicular to the bases. Find the volume of the obtained body of revolution.

Solution. Let's make a drawing (Fig. 37).

As a result of rotation, we get a truncated cone; in order to find its volume, we calculate the radius of the larger base and the height. in a trapeze O 1 O 2 AB we will spend AC^O 1 B. In we have: so this triangle is isosceles AC=BC\u003d 3 cm.

Answer:

Example 4 A triangle with sides 13 cm, 37 cm and 40 cm rotates around an external axis that is parallel to the larger side and is 3 cm away 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 side surfaces of two truncated cones and the side surface of the 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 OC) forming cones ( BC and AC) and the height of the cylinder ( AB). The unknown is only CO. is the distance from the side of the triangle to the axis of rotation. Let's find DC. The area of ​​triangle ABC on one side is equal to the product of half of side AB and the height drawn to it DC, on the other hand, knowing all the sides of the triangle, we calculate its area using Heron's formula.

- this is a part of a cone, limited between two parallel bases perpendicular to its axis of symmetry. The bases of the cone are geometric circles.

A truncated cone can be obtained by rotating a rectangular trapezoid around its side, which is its height. The boundary of the cone is a circle of radius R, a circle of radius r, and the lateral surface of the cone. The lateral surface of the cone describes the lateral side of the trapezoid during its rotation.

The area of ​​the lateral surface of a truncated cone through the guide and the radii of its bases

When finding the area side surface of a truncated cone, it is more expedient to consider it as the difference between the lateral surface of the cone and the lateral surface of the truncated cone.

Let the cone A`MB` be cut off from the given cone AMB. Need to calculate side area truncated cone AA`B`B . It is known that the radii of its bases are AO=R, A`O` =r , the generatrix is ​​equal to L . Let's denote MB` as x . Then the lateral surface of the cone A`MB` will be equal to πrx. And the lateral surface of the cone AMB will be equal to πR(L+x).
Then the lateral surface of the truncated cone AA`B`B can be expressed in terms of the difference between the lateral surface of the cone AMB and the cone A`MB` :

Triangles OMB and O`MB` are similar in terms of the equality of angles ∠(MOB) = ∠(MO`B`) and ∠(OMB) = ∠(O`MB`) . From the similarity of these triangles follows:
Let's use the derivative proportion. We have:
From here we find x :
Substituting this expression into the formula for the lateral surface area, we have:
Thus, the area of ​​the lateral surface of a truncated cone is equal to the product of the number π and its guide and the sum of the radii of its bases.

An example of calculating the area of ​​the lateral surface of a truncated cone, if its radius and generatrix are known
The radius of the larger base, the generatrix and the height of the truncated cone are 7, 5 and 4 cm, respectively. Find the area of ​​the lateral surface of the cone.
The axial section of a truncated cone is isosceles trapezoid, with bases 2R and 2r . The generatrix of the truncated cone, which is the lateral side of the trapezoid, the height, pubescent on the large base and the difference in the radii of the base of the truncated cone, form the Egyptian triangle. it right triangle with an aspect ratio of 3:4:5. According to the condition of the problem, the generatrix is ​​equal to 5, and the height is 4, then the difference in the radii of the base of the truncated cone will be equal to 3.
We have:
L=5
R=7
R=4
The formula for the area of ​​the lateral surface of a truncated cone is as follows:

Substituting the values, we have:

The area of ​​the lateral surface of the truncated cone through the guide and the average radius

The average radius of a truncated cone is equal to half the sum of the radii of its bases:


Then the formula for the area of ​​the lateral surface of a truncated cone can be represented as follows:

The area of ​​the lateral surface of a truncated cone is equal to the product of the circumference of the average section and its generatrix.

Areas of the lateral surface of a truncated cone through the radii of its base and the angle of inclination of the generatrix to the plane of the base

If the smaller base is orthogonally projected onto the larger base, then the projection of the lateral surface of the truncated cone will look like a ring, the area of ​​​​which is calculated by the formula:

Then:

Areas of the lateral surface of a truncated cone according to Archimedes

The area of ​​the lateral surface of a truncated cone is equal to the area of ​​such a circle, the radius of which is the average proportional between the generatrix and the sum of the radii of its bases

Full surface of a truncated cone

The total surface of a cone is the sum of the area of ​​its lateral surface and the area of ​​the bases of the cone:

The bases of the cone are circles with radius R and r. Their area is equal to the product of the number times the square of their radius:


The lateral surface area is calculated by the formula:

Then the total surface area of ​​the truncated cone is:

The formula looks like this:

An example of calculating the total surface area of ​​a truncated cone if its radius and generatrix are known
The radius of the base of the truncated cone is 1 and 7 dm, and the diagonals of the axial section are mutually perpendicular. Find the total area of ​​the truncated cone
The axial section of a truncated cone is an isosceles trapezoid, with bases 2R and 2r. That is, the bases of the trapezoid are 2 and 14 dm, respectively. Since the diagonals of a trapezoid are mutually perpendicular, the height is half the sum of its bases. Then:

The generatrix of the truncated cone, which is the lateral side of the trapezoid, the height, pubescent on the large base and the difference in the radii of the base of the truncated cone, form a right triangle.
By the Pythagorean theorem, we find the generatrix of a truncated cone:

The formula for the total surface area of ​​a truncated cone is:

Substituting the values ​​from the condition of the problem and the found values, we have:

Volume formulas

Truncated pyramid or cone - this is the part that remains after cutting off the top with a plane parallel to the base.

Volume of a truncated pyramid or cones equals the volume of the whole pyramid or cone minus the volume of the truncated vertex.

Lateral surface area of ​​a truncated pyramid or cones equal to the surface area of ​​an entire pyramid or cone. minus the side surface area of ​​the clipped vertex. If you need to find total area truncated figure, then the area of ​​the two parallel bases is added to the area of ​​the lateral surface.

There is another method for determining the volume and surface area of ​​a truncated cone:

V=1/3 π h(R 2 +Rr+r 2),

lateral surface area of ​​a cone S=πl(R+r),

total surface area S o \u003d π l (R + r) + πr 2 + πR 2

Example1. Determination of the area required for the manufacture of material for the lampshade. (Calculation of the area of ​​the lateral surface of the cone).

The lampshade has the shape of a truncated cone. The height of the lampshade is 50 cm, the lower and upper diameters are 40 and 20 cm, respectively.

Determine to within 3x significant figures the area of ​​the material needed to make the lampshade.

As defined above, the lateral surface area of ​​a truncated cone S=πl(R+r).

Since the upper and lower diameters of the truncated cone are 40 and 20 cm, from Fig. above we find r=10 cm, R=20 cm and

l \u003d (50 2 +10 2) 1/2 \u003d 50.99 according to the Pythagorean theorem,

Therefore, the area of ​​the lateral surface of the cone is S \u003d π 50.99 (20 + 10) \u003d 4803.258 cm 2, i.e. the area of ​​​​the material required for the manufacture of the lampshade is equal to 4800 cm2 accurate to 3 significant figures, although, of course, how much material will actually take depends on the cut.

Example 2. Determining the volume of a cylinder crowned with a truncated cone.

The cooling tower is shaped like a cylinder topped with a truncated cone, as shown in Fig. below. Determine the volume of airspace in the tower if 40% of the volume is occupied by pipes and other structures.

The volume of the cylindrical part

V=π R 2 h\u003d π (27/2) 2 * 14 \u003d 8011.71 m 3

Truncated Cone Volume

V=1/3 π h(R 2 +Rr+r 2), where

h=34-14=20 m, R=27/2=13.5 m and r=14/2=7 m.

Because R=27/2=13.5 m and r=14/2=7 m.

Therefore, the volume of the truncated cone

V \u003d 1/3 π 20 (13.5 2 + 13.5 * 7 + 7 2) \u003d 6819.03 m 3

Total Cooling Tower Volume V common. \u003d 6819.03 + 8011.71 \u003d 14830.74 m 3.

If 40% of the volume is occupied, airspace volume V \u003d 0.6 * 14830.74 \u003d 8898.44 m 3