In geometry, a truncated cone is a body that is formed by the rotation of a rectangular trapezoid about that side of it, which is perpendicular to the base. How do they calculate truncated cone volume, everyone knows from school course geometry, and in practice this knowledge is often used by designers of various machines and mechanisms, developers of some consumer goods, as well as architects.
Calculation of the volume of a truncated cone
The formula for calculating the volume of a truncated cone
The volume of a truncated cone is calculated by the formula:
| V | πh (R 2 + R × r + r 2) |
h- cone height
r- radius of the upper base
R- bottom base radius
V- volume of the truncated cone
π - 3,14
With such geometric bodies as truncated cones, V Everyday life everyone collides quite often, if not constantly. Their shape has a wide variety of containers widely used in everyday life: buckets, glasses, some cups. It goes without saying that the designers who developed them must have used a formula that calculates truncated cone volume, since this quantity has in this case a very great importance, because it determines such an important characteristic as the capacity of the product.
Engineering structures, which are truncated cones, often seen on large industrial enterprises, as well as thermal and nuclear power plants. It is this form that cooling towers have - devices designed to cool large volumes of water by forcing a counter flow of atmospheric air. Most often, these designs are used in cases where it is required to significantly reduce the temperature in a short time. a large number liquids. The developers of these structures must determine truncated cone volume the formula for calculating which is quite simple and known to all those who once studied well in high school.
Parts with this geometric shape, are often found in the design of various technical devices. For example, gears used in systems where it is required to change the direction of kinetic transmission are most often implemented using bevel gears. These parts are an integral part of a wide variety of gearboxes, as well as automatic and manual gearboxes used in modern cars.
The shape of a truncated cone has some cutting tools that are widely used in production, for example, milling cutters. With their help, you can process inclined surfaces at a certain angle. For sharpening cutters of metalworking and woodworking equipment, abrasive wheels are often used, which are also truncated cones. Besides, truncated cone volume it is required to determine the designers of turning and milling machines, which involve the fastening of a cutting tool equipped with tapered shanks (drills, reamers, etc.).
Sometimes the task arises - to make a protective umbrella for an exhaust or chimney, an exhaust deflector for ventilation, etc. But before you start manufacturing, you need to make a pattern (or scan) for the material. On the Internet there are all sorts of programs for calculating such sweeps. However, the problem is so easy to solve that you will quickly calculate it with a calculator (on a computer) than you will search, download and deal with these programs.
Let's start with a simple option - sweep simple cone. The easiest way to explain the principle of calculating the pattern is with an example.
Suppose we need to make a cone with a diameter of D cm and a height of H centimeters. It is quite clear that a circle with a cut segment will act as a blank. Two parameters are known - diameter and height. Using the Pythagorean theorem, we calculate the diameter of the workpiece circle (do not confuse it with the radius finished cones). Half the diameter (radius) and the height form a right triangle. That's why:
So, now we know the radius of the workpiece and we can cut out the circle.
Calculate the angle of the sector to be cut out of the circle. We argue as follows: The diameter of the workpiece is 2R, which means that the circumference is Pi * 2 * R - i.e. 6.28*R. We denote it by L. The circle is complete, i.e. 360 degrees. And the circumference of the finished cone is Pi * D. We denote it by Lm. It is, of course, less than the circumference of the workpiece. We need to cut a segment with an arc length equal to the difference between these lengths. Apply the ratio rule. If 360 degrees gives us the full circumference of the workpiece, then the desired angle should give the circumference of the finished cone.
From the ratio formula, we obtain the size of the angle X. And the cut sector is found by subtracting 360 - X.
From a round blank with a radius R, a sector with an angle (360-X) must be cut. Be sure to leave a small strip of overlapping material (if the cone mount will overlap). After connecting the sides of the cut sector, we get a cone of a given size.
For example: We need a chimney hood cone with a height (H) of 100 mm and a diameter (D) of 250 mm. According to the Pythagorean formula, we obtain the radius of the workpiece - 160 mm. And the circumference of the workpiece, respectively, 160 x 6.28 = 1005 mm. At the same time, the circumference of the cone we need is 250 x 3.14 = 785 mm.
Then we get that the ratio of angles will be: 785 / 1005 x 360 = 281 degrees. Accordingly, it is necessary to cut the sector 360 - 281 = 79 degrees.
Calculation of the pattern blank for a truncated cone.
Such a detail is sometimes needed in the manufacture of adapters from one diameter to another or for Volpert-Grigorovich or Khanzhenkov deflectors. They are used to improve draft in a chimney or ventilation pipe.
The task is slightly complicated by the fact that we do not know the height of the entire cone, but only its truncated part. In general, there are three initial numbers: the height of the truncated cone H, the diameter of the lower hole (base) D, and the diameter of the upper hole Dm (at the cross section of the full cone). But we will resort to the same simple mathematical constructions based on the Pythagorean theorem and similarity.
Indeed, it is obvious that the value (D-Dm) / 2 (half the difference in diameters) will relate with the height of the truncated cone H in the same way as the radius of the base to the height of the entire cone, as if it were not truncated. We find the total height (P) from this ratio.
(D – Dm)/ 2H = D/2P
Hence Р = D x H / (D-Dm).
Now knowing the total height of the cone, we can reduce the solution of the problem to the previous one. Calculate the development of the workpiece as if for a full cone, and then “subtract” from it the development of its upper, unnecessary part. And we can calculate directly the radii of the workpiece.
We obtain by the Pythagorean theorem a larger radius of the workpiece - Rz. This Square root from the sum of the squares of the heights P and D/2.
The smaller radius Rm is the square root of the sum of squares (P-H) and Dm/2.
The circumference of our workpiece is 2 x Pi x Rz, or 6.28 x Rz. And the circumference of the base of the cone is Pi x D, or 3.14 x D. The ratio of their lengths will give the ratio of the angles of the sectors, if we assume that the full angle in the workpiece is 360 degrees.
Those. X / 360 = 3.14 x D / 6.28 x Rz
Hence X \u003d 180 x D / Rz (This is the angle that must be left to get the circumference of the base). And you need to cut accordingly 360 - X.
For example: We need to make a truncated cone 250 mm high, base diameter 300 mm, top hole diameter 200 mm.
We find the height of the full cone P: 300 x 250 / (300 - 200) = 600 mm
According to the Pythagorean method, we find the outer radius of the workpiece Rz: The square root of (300/2) ^ 2 + 6002 = 618.5 mm
By the same theorem, we find the smaller radius Rm: The square root of (600 - 250)^2 + (200/2)^2 = 364 mm.
We determine the angle of the sector of our workpiece: 180 x 300 / 618.5 = 87.3 degrees.
On the material we draw an arc with a radius of 618.5 mm, then from the same center - an arc with a radius of 364 mm. The arc angle can have approximately 90-100 degrees of opening. We draw radii with an opening angle of 87.3 degrees. Our preparation is ready. Don't forget to allow for seam edges if they overlap.
Truncated Cone Definition
A truncated cone can be obtained from an ordinary cone if such a cone is intersected by a plane parallel to the base. Then the figure that is between two planes (this plane and the base of an ordinary cone) will be called a truncated cone.
He has two bases, which for a circular cone are circles, and one of them is larger than the other. The truncated cone also has height- a segment connecting two bases and perpendicular to each of them.
Online calculator
The truncated cone can be direct, then the center of one base is projected into the center of the second. If the cone inclined, then such a projection does not take place.
Consider a right circular cone. The volume of this figure can be calculated in several ways.
The formula for the volume of a truncated cone in terms of the radii of the bases and the distance between them
If we are given a circular truncated cone, then we can find its volume using the formula:
Truncated Cone VolumeV = 1 3 ⋅ π ⋅ h ⋅ (r 1 2 + r 1 ⋅ r 2 + r 2 2) V=\frac(1)(3)\cdot\pi\cdot h\cdot(r_1^2+r_1\ cdot r_2+r_2^2)V =3 1 ⋅ π ⋅ h ⋅(r 1 2 + r 1 ⋅ r 2 + r 2 2 )
R 1 , r 2 r_1, r_2 r 1
,
r 2
- radii of the bases of the cone;
h h h- the distance between these bases (the height of the truncated cone).
Consider an example.
Task 1Find the volume of a truncated cone if it is known that the area of the small base is 64 π cm 2 64\pi\text( cm)^26 4 pi cm2 , big - 169 π cm 2 169\pi\text( cm)^21 6 9 cm2 , and its height is 14 cm 14\text( cm) 1 4 cm.
Solution
S 1 \u003d 64 π S_1 \u003d 64 \ pi S 1
=
6 4 pi
S 2 \u003d 169 π S_2 \u003d 169 \ pi S 2
=
1 6 9
h=14 h=14 h =1
4
Find the radius of the small base:
S 1 = π ⋅ r 1 2 S_1=\pi\cdot r_1^2S 1 = π ⋅ r 1 2
64 π = π ⋅ r 1 2 64\pi=\pi\cdot r_1^26 4 π =π ⋅ r 1 2
64=r 1 2 64=r_1^2 6 4 = r 1 2
R1=8 r_1=8 r 1 = 8
Similarly, for the big base:
S 2 = π ⋅ r 2 2 S_2=\pi\cdot r_2^2S 2 = π ⋅ r 2 2
169 π = π ⋅ r 2 2 169\pi=\pi\cdot r_2^21 6 9π ⋅ r 2 2
169=r 2 2 169=r_2^2 1 6 9 = r 2 2
R2=13 r_2=13 r 2 = 1 3
Calculate the volume of the cone:
V = 1 3 ⋅ π ⋅ h ⋅ (r 1 2 + r 1 ⋅ r 2 + r 2 2) = 1 3 ⋅ π ⋅ 14 ⋅ (8 2 + 8 ⋅ 13 + 1 3 2) ≈ 4938 cm 3 V= \frac(1)(3)\cdot\pi\cdot h\cdot (r_1^2+r_1\cdot r_2+r_2^2)=\frac(1)(3)\cdot\pi\cdot14\cdot(8 ^2+8\cdot 13+13^2)\approx4938\text( cm)^3V =3 1 ⋅ π ⋅ h ⋅(r 1 2 + r 1 ⋅ r 2 + r 2 2 ) = 3 1 ⋅ π ⋅ 1 4 ⋅ (8 2 + 8 ⋅ 1 3 + 1 3 2 ) ≈ 4 9 3 8 cm3
Answer
4938 cm3. 4938\text(cm)^3.4 9 3 8 cm3 .
The formula for the volume of a truncated cone in terms of the areas of the bases and their distance to the top
Let's say we have a truncated cone. Mentally add the missing piece to it, thereby making it a “normal cone” with a vertex. Then the volume of a truncated cone can be found as the difference between the volumes of two cones with corresponding bases and their distance (height) to the top of the cone.
Truncated Cone VolumeV = 1 3 ⋅ S ⋅ H − 1 3 ⋅ s ⋅ h = 1 3 ⋅ (S ⋅ H − s ⋅ h) V=\frac(1)(3)\cdot S\cdot H-\frac(1) (3)\cdot s\cdot h=\frac(1)(3)\cdot (S\cdot H-s\cdot h)V =3 1 ⋅ S ⋅H-3 1 ⋅ s⋅h =3 1 ⋅ (S ⋅H-s⋅h)
S S S is the area of the base of the large cone;
HH H is the height of this (large) cone;
s s s- area of the base of the small cone;
h h h- the height of this (small) cone;
Determine the volume of the truncated cone if the height of the full cone is HH H is equal to 10 cm 10\text( cm)
Solution
R=5 R=5
Find the area of both bases of the cone:
S = π ⋅ R 2 = π ⋅ 5 2 ≈ 78.5 S=\pi\cdot R^2=\pi\cdot 5^2\approx78.5
s = π ⋅ r 2 = π ⋅ 4 2 ≈ 50.24 s=\pi\cdot r^2=\pi\cdot 4^2\approx50.24
Find the height of the small cone h h
H − h = 8 H-h=8
h = H − 8 h=H-8
h = 10 − 8 h=10-8
h=2 h=2
The volume is equal to the formula:
V = 1 3 ⋅ (S ⋅ H − s ⋅ h) ≈ 1 3 ⋅ (78.5 ⋅ 10 − 50.24 ⋅ 2) ≈ 228 cm 3 V=\frac(1)(3)\cdot (S\cdot H-s\cdot h)\approx\frac(1)(3)\cdot (78.5\cdot 10-50.24\cdot 2)\approx228\text( cm)^3
Answer
228 cm3. 228\text(cm)^3.
Instead of the word “pattern”, “sweep” is sometimes used, but this term is ambiguous: for example, a reamer is a tool for increasing the diameter of a hole, and in electronic technology there is a concept of a reamer. Therefore, although I am obliged to use the words “cone sweep” so that search engines can find this article using them, I will use the word “pattern”.
Building a pattern for a cone is a simple matter. Let us consider two cases: for a full cone and for a truncated one. On the picture (click to enlarge) sketches of such cones and their patterns are shown. (I note right away that we will only talk about straight cones with a round base. We will consider cones with an oval base and inclined cones in the following articles).
1. Full taper
Designations:
Pattern parameters are calculated by the formulas:
;
;
Where 
.
2. Truncated cone
Designations:
Formulas for calculating pattern parameters: 
;
;
;
Where 
.
Note that these formulas are also suitable for the full cone if we substitute .
Sometimes, when constructing a cone, the value of the angle at its vertex (or at the imaginary vertex, if the cone is truncated) is fundamental. The simplest example is when you need one cone to fit snugly into another. Let's denote this angle with a letter (see picture).
In this case, we can use it instead of one of the three input values: , or . Why "together O", not "together e"? Because three parameters are enough to construct a cone, and the value of the fourth is calculated through the values of the other three. Why exactly three, and not two or four, is a question that is beyond the scope of this article. A mysterious voice tells me that this is somehow connected with the three-dimensionality of the “cone” object. (Compare with the two initial parameters of the two-dimensional circle segment object, from which we calculated all its other parameters in the article.)
Below are the formulas by which the fourth parameter of the cone is determined when three are given.
4. Methods for constructing a pattern
- Calculate the values on the calculator and build a pattern on paper (or immediately on metal) using a compass, ruler and protractor.
- Enter formulas and source data into a spreadsheet (for example, Microsoft Excel). The result obtained is used to build a pattern using a graphic editor (for example, CorelDRAW).
- use my program, which will draw on the screen and print out a pattern for a cone with given parameters. This pattern can be saved as a vector file and imported into CorelDRAW.
5. Not parallel bases
As far as truncated cones are concerned, the Cones program still builds patterns for cones that have only parallel bases.
For those who are looking for a way to construct a truncated cone pattern with non-parallel bases, here is a link provided by one of the site visitors:
A truncated cone with non-parallel bases.
Geometry as a science was formed in Ancient Egypt and reached high level development. The famous philosopher Plato founded the Academy, where close attention was paid to the systematization of existing knowledge. The cone as one of the geometric figures was first mentioned in the famous treatise of Euclid "Beginnings". Euclid was familiar with the works of Plato. Now few people know that the word "cone" is translated from Greek means "pine cone". The Greek mathematician Euclid, who lived in Alexandria, is rightfully considered the founder of geometric algebra. The ancient Greeks not only became the successors of the knowledge of the Egyptians, but also significantly expanded the theory.
History of the definition of a cone
Geometry as a science emerged from practical requirements construction and observation of nature. Gradually, experimental knowledge was generalized, and the properties of some bodies were proved through others. The ancient Greeks introduced the concept of axioms and proofs. An axiom is a statement obtained in a practical way and does not require proof.
In his book, Euclid gave the definition of a cone as a figure that is obtained by rotation right triangle around one of the legs. He also owns the main theorem that determines the volume of a cone. And the ancient Greek mathematician Eudoxus of Cnidus proved this theorem.
Another mathematician ancient greece, Apollonius of Perga, who was a student of Euclid, developed and expounded the theory of conic surfaces in his books. He owns the definition of a conical surface and a secant to it. Schoolchildren of our days are studying Euclidean geometry, which has preserved the main theorems and definitions from ancient times.
Basic definitions
A right circular cone is formed by the rotation of a right triangle around one leg. As you can see, the concept of a cone has not changed since the time of Euclid.
The hypotenuse AS of a right triangle AOS, when rotating around the leg OS, forms side surface of a cone and is therefore called a generatrix. The leg OS of the triangle turns simultaneously into the height of the cone and its axis. Point S becomes the apex of the cone. The leg AO, having described the circle (base), turned into the radius of the cone.
If we draw a plane from above through the vertex and axis of the cone, we can see that the resulting axial section is an isosceles triangle, in which the axis is the height of the triangle.
Where C- base circumference, l is the length of the generatrix of the cone, R is the radius of the base.
The formula for calculating the volume of a cone
The following formula is used to calculate the volume of a cone:
where S is the area of the base of the cone. Since the base is a circle, its area is calculated as follows:
This implies:
where V is the volume of the cone;
n is a number equal to 3.14;
R is the radius of the base corresponding to the segment AO in Figure 1;
H is the height equal to the segment OS.
Truncated cone, volume
There is a right circular cone. If the upper part is cut off by a plane perpendicular to the height, then a truncated cone will be obtained. Its two bases have the shape of a circle with radii R 1 and R 2 .
If a right cone is formed by the rotation of a right triangle, then a truncated cone is formed by the rotation of a right-angled trapezoid around the straight side.
The volume of a truncated cone is calculated using the following formula:
V \u003d n * (R 1 2 + R 2 2 + R 1 * R 2) * H / 3.
Cone and its section by a plane
Peru of the ancient Greek mathematician Apollonius of Perga belongs to the theoretical work "Conic Sections". Thanks to his work in geometry, definitions of curves appeared: parabola, ellipse, hyperbola. Consider, and here the cone.
Take a right circular cone. If the plane intersects it perpendicular to the axis, then a circle is formed in the section. When the secant crosses the cone at an angle to the axis, then an ellipse is obtained in the section.
The secant plane, perpendicular to the base and parallel to the axis of the cone, forms a hyperbola on the surface. A plane cutting the cone at an angle to the base and parallel to the tangent to the cone creates a curve on the surface, which is called a parabola.
The solution of the problem
Even simple task how to make a bucket of a certain volume requires knowledge. For example, you need to calculate the dimensions of a bucket so that it has a volume of 10 liters.
V \u003d 10 l \u003d 10 dm 3;
The development of the cone has the form shown schematically in Figure 3.
L - generatrix of the cone.
To find out the surface area of a bucket, which is calculated using the following formula:
S \u003d n * (R 1 + R 2) * L,
it is necessary to calculate the generatrix. We find it from the volume value V \u003d n * (R 1 2 + R 2 2 + R 1 * R 2) * H / 3.
Hence H=3V/n*(R 1 2 +R 2 2 +R 1 *R 2).
A truncated cone is formed by rotating a rectangular trapezoid, in which the lateral side is the generatrix of the cone.
L 2 \u003d (R 2- R 1) 2 + H 2.
Now we have all the data to build the bucket drawing.
Why are fire buckets shaped like a cone?
Who wondered why fire buckets have a seemingly strange conical shape? And it's not just that. It turns out that when extinguishing a fire, a conical bucket has many advantages over a conventional, truncated cone-shaped one.
Firstly, as it turns out, the fire bucket fills with water faster and does not spill when carried. A cone larger than a regular bucket allows you to carry more water at a time.
Secondly, water from it can be thrown out to a greater distance than from a conventional bucket.
Thirdly, if the conical bucket falls off the hands and falls into the fire, then all the water is poured onto the fire.
All of these factors save time - the main factor in extinguishing a fire.
Practical use
Schoolchildren often have the question of why to learn how to calculate the volume of various geometric bodies, including a cone.
And design engineers are constantly faced with the need to calculate the volume of the conical parts of mechanism parts. These are the tips of drills, parts of turning and milling machines. The shape of the cone will allow the drills to easily enter the material without requiring initial basting with a special tool.
The volume of the cone has a pile of sand or earth poured onto the ground. If necessary, by making simple measurements, you can calculate its volume. For some, the question of how to find out the radius and height of a pile of sand will cause difficulty. Armed with a tape measure, we measure the circumference of the mound C. Using the formula R \u003d C / 2n, we find out the radius. Throwing a rope (roulette) over the top, we find the length of the generatrix. And to calculate the height using the Pythagorean theorem and volume is not difficult. Of course, such a calculation is approximate, but it allows you to determine if you were not deceived by bringing a ton of sand instead of a cube.
Some buildings are shaped like a truncated cone. For example, the Ostankino television tower is approaching the shape of a cone. It can be represented as consisting of two cones placed on top of each other. The domes of ancient castles and cathedrals are a cone, the volume of which the ancient architects calculated with amazing accuracy.
If you look closely at the surrounding objects, then many of them are cones:
- funnels for pouring liquids;
- horn-loudspeaker;
- parking cones;
- lampshade for floor lamp;
- the usual Christmas tree;
- wind musical instruments.
As can be seen from the above examples, the ability to calculate the volume of a cone, its surface area is necessary in professional and everyday life. We hope this article will help you.
