A cone is a three-dimensional rotation figure that often appears in geometric problems. Like any figure, it has a number of linear characteristics, surface area and volume. In this article, we consider what a cone is, and also give the formula for the generatrix of the cone.
Geometry cone
Many ate an ice cream cone on a hot summer day. The waffle cup of this ice cream is cone-shaped. In geometry, the stricter definition of this figure is as follows: a cone is a surface that is obtained by connecting, using straight segments, all points of a plane curve with a fixed point in space that does not lie in the plane of the curve. If the curve is an ellipse, then the cone will be elliptical, if the circle is round, and so on.
The geometric definition given above corresponds to the figure in the figure below.
Here, the curve on which the conical surface rests is the circle whose center lies on the vertical axis. As can be seen from the figure, although the figure is spatial, but its volume is zero. If instead of a circle we take a circle, then the volume of space bounded by this circle and a conical surface will be the volume of the cone.
Shape Elements
Before giving the formula for the generatrix of the cone, it should be clarified what elements the spatial figure under consideration consists of and what geometric concepts are used to describe it.
Based on the previous figure, we can say that the cone is formed by two surfaces:
- conical, which is also called lateral;
- a flat figure (the circle in the figure), which is called the base of the cone.
Unlike polyhedra, the cone does not have faces and edges, however, it has one single vertex, which borders the conical surface from above. In the previous figure, the vertex is at the origin.
For the considered figure, the concepts of generatrix and guide are also used. What is the generatrix of the cone, and what is its guide. Let's start by defining the latter. The guideline is a curve that limits the base of the shape. It is called so, because a generatrix moves along it, describing a conical surface. A guide is often called a director.
A generatrix is a straight segment that connects the top of the figure and any point on the guide. There are an infinite number of generators in a cone. Their combination defines a conical surface. The generator is also called generatrix.
Round straight cone
As noted above, the base of the cone can be bounded by any plane curve. Nevertheless, in practice and in geometric problems, a round cone is often found. We will consider it further in the article.
The round cone can be straight or inclined. In the first case, the perpendicular drawn from its top to the base intersects the latter in the center of the circle. A straight circular cone is a symmetrical figure of rotation, which can be obtained using a rectangular triangle. The figure below shows how to do this.
The figure shows a right triangle that stands on one of its legs. Around the other leg, which is the axis, a flat figure rotates. During rotation, the hypotenuse of a triangle describes a conical surface.
Linear characteristics of a round straight cone
Below is a drawing of a round cone, which introduced the notation. The value h is the height of the figure, r is the radius of the round base, which is called the radius of the cone, s is the length of the generatrix.
Obviously, all generators s of the considered figure have the same length. This important property is inherent only in a straight round cone. The angle between any generator and the axis of rotation of the cone is a constant at known values of r and h.
As you can see from the figure, the sides r, h and s bound a right triangle. This fact allows us to apply the Pythagorean theorem to write mathematical equality relating the named linear characteristics of the cone. Equality takes the form:
s 2 = r 2 + h 2
This equality contains the answer to the question of how to find the length of the generatrix of the cone. If the radius of the figure and its height are known, then:
s = √ (r 2 + h 2 )
Further, by examples of solving problems, we show how to use the formula of the generator of the cone.
The problem with the area of the conical surface and the circumference of the base
A straight cone with a round base is given. It is known that the area of its lateral surface is 120 cm 2 and the circumference of the base is 30 cm. What is the generatrix of the cone?
The solution to this problem involves the use of two additional formulas that were not considered in the article. We give them below:
S b = pi × r × s;
L = 2 × pi × r
The first expression is a formula for calculating the area of a conical surface S b . The second expression is familiar to every student, this is a formula for determining the circumference L.
Express the radius r from the second expression, substitute it into the first:
r = L / (2 × pi);
S b = pi × L / (2 × pi) × s
Then the desired formula for the length of the generatrix of the cone will take the form:
s = 2 × S b / L
We substitute the data of area S b and length L in this formula, we get the answer: s = 8 cm.
Task with base area and height
It is known that the round base of the cone has an area of 40 cm 2 . The height of the figure is 2 times the diameter of this base. It is necessary to find the length of the generatrix of the cone.
As in the previous case, we write two equations based on the condition of the problem:
S o = pi × r 2 ;
h = 4 × r
The first formula is a well-known expression for the area of a circle. The second equality follows from the fact that the height is 2 times larger than the diameter, which means 4 times the radius r.
Both expressions allow expressing r and h through S o :
r = √ (S o / pi);
h = 4 × √ (S o / pi)
Now we use the formula of the generator of the cone in terms of h and r. We get:
s = √ (S o / pi + 16 × S o / pi) = √ (17 × S o / pi)
We substitute the value of S o into equality and write the answer: s ≈ 14.72 cm.