The cylinder and cone are rotation figures, which are obtained by moving the generatrix along the guide line. In this article, we will reveal the question that this is a generatrix of a cone, and also show how to calculate its length in the case of a conventional cone and its truncated version.
General cone shape
We are talking about a geometric volumetric figure, which is obtained as follows: imagine that there is some curve in the plane, for example, an ellipse. We choose in space absolutely any point that does not lie in the plane of the ellipse. Now, using a straight segment, connect the point and the ellipse, and then we will move this segment along the elliptic curve. The result of this movement will be a conical surface. It is shown below in the figure.
Here, the point at the origin is called the vertex of the cone. An elliptical curve is a guide or director of a shape. A cone generator is a line that connects its vertex to any point on the guide curve. Generator is also called generatrix. From the figure it is clear that the length of the generatrix changes as it moves along the ellipse.
Round cone
Round is any cone whose guiding curve is a circle. If the perpendicular, which is drawn from the top of this cone, intersects the circular base in the center, then such a figure will be called a straight line. A round straight cone is shown in the figure.
The figure clearly shows that for this figure, the height h, any radius r, and the generator s make up a right triangle. Using abstract imagination, we can imagine that if this triangle is rotated around h, then we get a round cone.
The cone generator is the hypotenuse of the described right triangle. This fact allows us to write down the equality that can be used to calculate the length of the generator s:
s = β (r 2 + h 2 )
Note that, in contrast to the general type cone, all the lengths of the generators for a round figure are equal to each other and make up a certain angle with the base. If we denote this angle by the letter Ο, then we can write two more formulas for determining the length s:
s = r / cos (Ο);
s = h / sin (Ο).
Truncated round cone
Suppose we have a trapezoid with two right angles. If you rotate it around the side adjacent to these corners, you get a truncated cone.
The figure shows the trapezoid in dashed lines. Here r and R are the radii of the small and large bases, h is the height, g is the generatrix of the truncated cone. The figure shows that all the generators for this figure are the same. To determine the length g, the following three formulas can be written:
g = β ((Rr) 2 + h 2 );
g = h / sin (Ο);
g = (Rr) / cos (Ο).
Here, the angle Ο corresponds to the angle between the generatrix and the large base.
Note that a truncated cone can be obtained from a conventional cone, if cut off by the plane that is parallel to its base, the upper part of the figure.
The problem of determining the truncated cone generatrix
We solve an interesting geometric problem. Suppose there is a truncated round cone. It is known that the height of this figure is 14 cm, and the angle between its generatrix and the small base is 135 o . It is necessary to find the length of the generatrix of the cone.
If the angle between the generatrix and the large base were given, then we could immediately apply the corresponding formula to calculate g. Nevertheless, this angle Ο is not difficult to determine.
Refer to the previous figure. The angle between r and g is known. If you move the height h along the radius r to its end, then the sides h, Rr and g form a right triangle. Since the angle between h and r is 90 o , then between h and g it will be:
Ξ± = 135 o - 90 o = 45 o .
This means that the angle Ο will also be equal to 45 o , since it is the angle of the right triangle mentioned above.
To determine the length of the generatrix g, substitute the known data in the formula. We get:
g = h / sin (Ο) = 14 / sin (45 o ) β 19.8 cm.
For any cone, the length of its generatrix is ββgreater than the height.