How to find the generatrix of a regular and truncated cone. Formulas

Spatial figures are examined in detail in the upper grades of secondary schools in the course of stereometry. This article answers the question of how to find the generatrix of a cone of a round straight line and the generatrix of the corresponding truncated figure.

Cone figure

To understand how to find the generatrix of the cone, you should give an idea of ​​this figure. A round straight cone is a figure of rotation of a rectangular triangle around one of its legs. The figure below shows the rotation process.

The resulting spatial figure has the following characteristics:

1. The side AB of the triangle is the height h of the cone. It lies on the axis of rotation of the figure.
2. The side AC of the triangle is the radius r of the cone. The circle that describes this radius is called the base of the shape.
3. The side CB of the triangle for the cone is its generatrix, or generatrix. She got this name because in the process of rotation she describes a conical surface.
4. The vertex B of the triangle is the vertex of the cone.

Note that the height of the figure intersects the circular base in its center. This is a sufficient condition to consider the cone straight.

Cone forming

Now we can proceed to answer the question of how to find the generatrix of the cone of a round straight line. It was said above that it is a segment that lies on a conical surface and connects the vertex to the point of the circle of the base. In the right-angled triangle from which the cone was derived, the generatrix is ​​hypotenuse. This observation allows us to write down the well-known Pythagorean theorem by relating the generator g with radius r and height h of the figure. The formula, how to find the generatrix of the cone, has the form:

g = √ (r ² + h ² )

In addition to this formula, in practice, instead of the height or radius of the figure, the angle φ between the generatrix and the base can be known. In this case, the generatrix g can be calculated using the following expressions:

g = h / sin (φ);

g = r / cos (φ)

These formulas follow from the properties of the trigonometric functions of the sine and cosine.

Thus, the calculation of the generatrix of the cone is possible if you know any two parameters of the figure.

Truncated cone shape

It is also a rotation figure, only instead of a right triangle it is necessary to rotate a rectangular trapezoid. The figure below shows a truncated cone.

Here the blue arrows indicate a rectangular trapezoid. The length of the vertical arrow is the height h of the figure, the lengths of the other two blue arrows are the radii of the base of the cone. Unlike a cylinder, the bases of a truncated cone have a different area. Denote their radii r ₁ and r ₂ . The fourth side of the trapezoid, inclined to the base, is a generatrix or generatrix. As for a regular cone, for a truncated one, all generatrices are equal to each other and form the side surface of the figure.

Note that the truncated cone received this name because it can be obtained not only by rotating the trapezoid, but also by cutting off the top of the round straight cone by the plane.

Truncated Generator

So, we met with a truncated cone, as well as with the concept of its generatrix. How to find the generatrix of a truncated cone? In order to obtain the desired formula, we note that if the height h is transferred parallel to itself to the side surface of the cone so that it touches one end of the generatrix, then we get a right triangle. Its sides are the height h (leg), the generator g (hypotenuse) and r ₁ -r ₂ (leg). Then we can write the formula for determining g:

g = √ ((r ₁ - r ₂ ) ² + h ² )

Accordingly, if an acute angle φ _{1 is given} between the large base and the generatrix, then the latter can be determined as follows:

g = h / sin (φ ₁ );

g = (r ₁ - r ₂ ) / cos (φ ₁ )

If the obtuse angle φ ₂ between the small base and the generatrix is ​​known, then for its calculation it is necessary to use the following expressions:

g = h / sin (φ ₂ );

g = (r ₂ - r ₁ ) / cos (φ ₂ )

Here, the first formula is exactly the same as for the angle φ ₁ , and in the second formula, the radii in the numerator are swapped.

Thus, it is possible to find the generator of the truncated cone if you know any three of its parameters.