Consideration of the properties of volumetric figures is one of the priority problems of geometry. Important characteristics of all spatial figures are volume and surface area. The article reveals the question that this is a truncated cone, and formulas are given for determining its surface area and volume.
Cone figure
Before answering the question that this is a truncated cone, you need to get acquainted with the figure from which it is formed. In geometry, it is customary to call a cone a figure obtained by connecting straight segments of a point in space with all points of a given curve. The point of space should not lie in the plane of the curve, it is called the vertex of the cone. The connecting segments are the forming figures, and the flat curve is the guideline. It borders the base of the cone. In turn, the totality of all generators is called a conical surface. The cone, the base of which is a circle, is shown in the figure.
The distance between the top of the figure and the base is called height. If the corresponding perpendicular intersects the base at the geometric center, then the figure is called a straight line.
Further in the article we will show how, using a straight round cone, to get a truncated figure.
Truncated cone and methods for its preparation
Suppose we have a figure that was shown in the previous paragraph. Take a plane parallel to the base of the cone and cut off the vertex of the figure with it. This process is shown in the figure.
The figure formed above the plane is a cone, but the figure under the plane is a truncated cone.
There is another way to obtain the figure in question. Suppose that there is some trapezoid with two right angles. If you rotate this trapezoid around the side to which the right angles are adjacent, then it will describe the surface of the truncated cone. This method of obtaining the figure is shown in the diagram below.
The side of the trapezoid around which the rotation was performed will be the axis of the truncated cone. The segment that the two bases of the figure cut off on the axis is called the height. The figure shows the generatrix g and the radii of the bases of the cone of truncated r and r '.
Finally, the third way to obtain a truncated cone is to increase the number of edges of the truncated pyramid to an infinite number. During this process, the pyramid will gradually turn into a cone.
It is curious to note that the shape of the considered geometric figure in a first approximation in nature is characteristic of an active volcano, which is clearly visible in the next photo.
Elements of a figure and its linear characteristics
A truncated cone is a spatial shape consisting of three surfaces. Two of them are round bases (upper and lower) and one is the side surface. Unlike polyhedrons, the figure in question has no vertices and faces.
Important parameters of the truncated cone are the radii of each of the bases. We will denote the larger radius r 1 , the smaller - r 2 . In addition to the radii of the figure, for its unambiguous determination, it is necessary to know either the height h or the generator g. The indicated parameters are mathematically related by the following equality:
g 2 = h 2 + (r 1 - r 2 ) 2
All four parameters are used to determine surface area and volume.
Truncated cone surface
As noted, the surface of the figure consists of three parts. If we cut off each of the bases from the figure, and then cut and expand the side surface along the generatrix, then we get a scan of a truncated cone. The figure below shows how it looks.
The base areas of the truncated cone are found by a simple formula for the corresponding circle size:
S o1 = pi × r 1 2 ;
S o2 = pi × r 2 2
With the area of the lateral surface, the situation is somewhat more complicated. It can be noted that it represents a sector of a circle of a certain radius G, from which the central part with a radius Gg was cut out. If we take this into account, then we can obtain the formula for the lateral surface area S b . Here we confine ourselves to the reduction of the final expression:
S b = pi × (r 1 + r 2 ) × g
This expression can be written in terms of radii and height h, but then it will have a somewhat cumbersome appearance.
Adding the written expressions, we obtain a formula for determining the area S of the entire surface of a truncated cone:
S = S o1 + S o2 + S b = pi × r 1 2 + pi × r 2 2 + pi × (r 1 + r 2 ) × g =
= pi × (r 1 2 + r 2 2 + (r 1 + r 2 ) × g)
Figure volume
Like any figure in space, a truncated cone also has some volume. This volume is limited by two bases and a lateral surface. Here we will not give a detailed conclusion of the corresponding formula for V. We write, as in the case with the surface area, only the final result:
V = h × pi / 3 × (r 1 2 + r 2 2 + r 1 × r 2 )
This formula, in contrast to the expression for the area S, contains, as parameters, the radii of the truncated cone and its height.
Further in the article we show how to use the above formulas to solve a specific geometric problem.
The task of determining the area and volume of a truncated cone
The figure below shows a truncated cone and its linear parameters. Necessary to find the surface area and volume of the figure.
We begin to solve the problem by determining the value V. Its calculation is not difficult, since all the necessary parameters are known. Substituting them in the formula for V, we get:
V = h × pi / 3 × (r 1 2 + r 2 2 + r 1 × r 2 ) =
= 10 × 3.14 / 3 × (8 2 + 1 2 + 8 × 1) ≈ 764.07 cm 3
The found value corresponds to 0.76 liters.
To find the surface area S, you must first calculate the length of the generatrix g of the figure. It will be equal to:
g = √ (h 2 + (r 1 - r 2 ) 2 ) = √ (10 2 + (8 - 1) 2 ) ≈ 12.21 cm
We rounded the value of the generator g to the hundredths. Now you can use the formula for area S:
S = pi × (r 1 2 + r 2 2 + (r 1 + r 2 ) × g) = 3.14 × (8 2 + 1 2 + (8 + 1) × 12.21) ≈ 549.15 cm 2
Note that the formulas for V and S that we used to solve the problem are valid only for a round straight truncated cone. In the case of an inclined figure or a figure with non-circular bases, these formulas cannot be used.