Volume formulas

Volumes and surface areas of truncated pyramids and cones.

Truncated pyramid or cone - this is the part remaining after the top is cut off by a plane parallel to the base.

Volume of a truncated pyramid or cone equal to the volume of the whole pyramid or cone minus the volume of the cut off vertex.

Lateral surface area of ​​a truncated pyramid or cone equal to the surface area of ​​the entire pyramid or cone. minus the lateral surface area of ​​the cut off vertex. If it is necessary to find the total area of ​​a truncated figure, then the area of ​​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 ​​the cone S=π l(R+r),

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

Example 1. Determining the area required for the manufacture of material for the lampshade. (Calculation of the lateral surface area 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 with an accuracy of 3x significant figures the area of ​​material required to make the lampshade.

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

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

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

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

Example 2. Determination of the volume of a cylinder topped by a truncated cone.

The cooler tower has the shape of a cylinder topped with a truncated cone, as shown in Fig. below. Determine the volume of air space in the tower if 40% of the volume is occupied by pipes and other structures.

Volume of the cylindrical part

V=π R 2 h=π(27/2) 2 *14=8011.71 m 3

Volume of a truncated cone

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 a truncated cone

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

Total volume of cooling tower V total =6819.03+8011.71=14830.74 m3.

If 40% of the volume is occupied, air space volume V=0.6*14830.74=8898.44 m 3

Cone. truncated cone

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 , point – 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).

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 its 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).

A truncated cone can be considered as a body obtained by rotation 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, then 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. Rectangular trapezoid with acute angle 45 O, with 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 rotation.

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.

- this is a part of a cone, bounded 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 is described by 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 lateral surface It is more appropriate to consider the truncated cone as the difference between the lateral surface of the cone and the lateral surface of the cut off cone.

Let the cone A`MB` be cut off from a given cone AMB. Need to calculate lateral area truncated cone AA`B`B. It is known that the radii of its bases are AO=R, A`O` =r, the generator is equal to L. Let us 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 through 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 equality of angles ∠(MOB) = ∠(MO`B`) and ∠(OMB) = ∠(O`MB`) . From the similarity of these triangles it follows:
Let's use the derivative of the proportion. We have:
From here we find x:
Substituting this expression into the lateral surface area formula, we have:
Thus, the area of ​​the lateral surface of a truncated cone is equal to the product of the number π by its guide and the sum of the radii of its bases.

An example of calculating the lateral surface area of ​​a truncated cone if its radius and generatrix are known
The radius of the larger base, the generator and the height of the truncated cone are 7, 5 and 4 cm, respectively. Find the lateral surface area 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 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 an Egyptian triangle. It is a right triangle with an aspect ratio of 3:4:5. According to the conditions 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 lateral surface area of ​​a truncated cone is as follows:

Substituting the values, we have:

Lateral surface area of ​​a truncated cone through a guide and 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 presented as follows:

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

Area 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 larger base, then the projection of the lateral surface of the truncated cone will have the form of a ring, the area of ​​which is calculated by the formula:

Then:

Lateral surface area of ​​a truncated cone according to Archimedes

The area of ​​the lateral surface of a truncated cone is equal to the area of ​​a circle whose radius 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 radii 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 ​​a truncated cone
The axial section of the 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 equal to half the sum of its bases. Then:

The generatrix of the truncated cone, which is the 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.
Using the Pythagorean theorem, we find the generatrix of a truncated cone:

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

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