One of the figures that occurs in solving geometric problems in space is a cone. It, unlike polyhedrons, belongs to the class of rotation figures. We consider in the article what is meant by it in geometry, and examine the characteristics of various sections of the cone.
Geometry cone
Suppose that there is some curve in the plane. It can be a parabola, a circle, an ellipse, and so on. Take a point that does not belong to the indicated plane, and connect all points of the curve with it. The formed surface is called a conical or simply cone.
If the original curve is closed, then the conical surface can be filled with matter. The figure thus obtained is a three-dimensional body. It is also called a cone. Several cones made of paper are shown in the figure below.
The conical surface is found in ordinary life. For example, this form has an ice cream cone or a road striped cone, which is designed to attract the attention of drivers and pedestrians.
Types of cones
As you might guess, the figures in question differ from each other by the type of curve on which they are formed. For example, there is a round cone or elliptical. This curve is called the base of the figure. However, the shape of the base is not the only feature that allows classification of cones.
Their second important characteristic is the position of the height relative to the base. The height of the cone is called a straight segment, which is lowered from the top of the figure to the plane of the base and is perpendicular to this plane. If the height intersects the base in the geometric center (for example, in the center of the circle), then the cone will be straight, if the perpendicular segment falls to any other point of the base or beyond, then the figure will be inclined.
Further in the article we will consider only a round straight cone as a bright representative of the considered class of figures.
Geometric names of cone elements.
It was said above that the cone has a base. It is bounded by a circle called a cone guide. The segments connecting the guide with a point that does not lie in the plane of the base are called generators. The set of all points of the generators is called the conical or lateral surface of the figure. For a round straight cone, all generators have the same length.
The point where the generators intersect is called the top of the figure. Unlike polyhedrons, a cone has a single vertex and does not have a single face.
A straight line passing through the top of the figure and the center of the circle is called the axis. The axis contains the height of the straight cone, so it forms a right angle with the plane of the base. This information is important in calculating the axial sectional area of a cone.
Round straight cone - rotation figure
The cone under consideration is a fairly symmetrical figure, which can be obtained as a result of the rotation of the triangle. Suppose there is a triangle with a right angle. To get a cone, it is enough to rotate this triangle around one of the legs as shown in the figure below.
It can be seen that the axis of rotation is the axis of the cone. One of the legs will be equal to the height of the figure, and the second leg will become the radius of the base. The hypotenuse of the triangle as a result of rotation will describe the conical surface. It will be a generatrix of the cone.
This method of obtaining a round straight cone is conveniently used to study the mathematical relationship between the linear parameters of the figure: height h, radius of the circular base r and the guide g. The corresponding formula follows from the properties of a right triangle. It is given below:
g 2 = h 2 + r 2 .
Since we have one equation and three variables, this means that to uniquely specify the parameters of a round cone, it is necessary to know any two quantities.
Cone sections by a plane that does not contain a vertex of the figure
The question of constructing sections of a figure is not trivial. The fact is that the shape of the cross section of the cone by the surface depends on the relative position of the figure and the secant.
Suppose we intersect a cone with a plane. What section will result from this geometric operation? Sectional shape options are shown in the figure below.
The pink section is a circle. It is formed as a result of the intersection of the figure with a plane that is parallel to the base of the cone. This section is perpendicular to the axis of the figure. The figure formed above the secant plane is a cone similar to the original one, but having a smaller circle at the base.
The green section is an ellipse. It is obtained if the secant plane is not parallel to the base, however, it intersects only the side surface of the cone. The figure cut off above the plane is called an elliptical inclined cone.
The blue and orange sections are parabolic and hyperbolic, respectively. As can be seen from the figure, they are obtained if the secant plane simultaneously intersects the side surface and the base of the figure.
To determine the cross-sectional areas of the cone that were considered, it is necessary to use formulas for the corresponding figure on the plane. For example, for a circle, this is the number of Pi multiplied by the square of the radius, and for an ellipse, this is the product of Pi by the length of the minor and major axes:
circle: S = pi * r 2 ;
ellipse: S = pi * a * b.
Sections containing a vertex of a cone
Now consider the options for the cross sections that arise if the secant plane passes through the top of the cone. Three cases are possible:
- Section is the only point. For example, a plane passing through a vertex and parallel to the base gives just such a section.
- The cross section is straight. This situation occurs when the plane is tangent to the conical surface. The direct section in this case will be the generatrix of the cone.
- Axial section. It is formed when the plane contains not only the top of the figure, but also its entire axis. In this case, the plane will be perpendicular to the round base and divide the cone into two equal parts.
Obviously, the areas of the first two types of sections are zero. As for the cone cross-sectional area for the 3rd species, this question is considered in more detail in the next paragraph.
Axial section
It was noted above that the axial section of a cone is a figure formed at the intersection of a cone by a plane passing through its axis. It is easy to guess that this section will represent the figure shown in the figure below.
This is an isosceles triangle. The top of the axial section of the cone is the top of this triangle, formed by the intersection of the same sides. The latter are equal to the length of the generatrix of the cone. The base of the triangle is the diameter of the base of the cone.
The calculation of the area of the axial section of the cone is reduced to finding the area of the resulting triangle. If the radius of the base r and the height h of the cone are initially known, then the area S of the considered section will be equal to:
S = h * r.
This expression is the result of applying the standard formula for the area of a triangle (half the product of the height and the base).
Note that if the generatrix of the cone is equal to the diameter of its round base, then the axial section of the cone is an equilateral triangle.
A triangular section is formed when the secant plane is perpendicular to the base of the cone and passes through its axis. Any other plane parallel to the named will give a hyperbole in cross section. However, if the plane contains the vertex of the cone and does not cross its base through the diameter, then the resulting section will also be an isosceles triangle.
The problem of determining the linear parameters of the cone
We show how to use the formula written for the axial section area for solving a geometric problem.
It is known that the axial sectional area of the cone is 100 cm 2 . The triangle obtained in the section is equilateral. What is the height of the cone and the radius of its base?
Since the triangle is equilateral, its height h is connected with the length of the side a by the following relation:
h = √3 / 2 * a.
Given that the side of the triangle is two times the radius of the base of the cone, and substituting this expression in the formula for the cross-sectional area, we obtain:
S = h * r = √3 / 2 * 2 * r * r =>
r = √ (S / √3).
Then the height of the cone is equal to:
h = √3 / 2 * 2 * r = √3 * √ (S / √3) = √ (√3 * S).
It remains to substitute the area value from the conditions of the problem and get the answer:
r = √ (100 / √3) ≈ 7.60 cm;
h = √ (√3 * 100) ≈ 13.16 cm.
In what areas is it important to know the parameters of the considered sections?
The study of various types of cone sections is not only of theoretical interest, but also has a practical application.
Firstly, it should be noted the area of aerodynamics, where using conical sections it is possible to create ideal smooth forms of solids.
Secondly, conical sections are the trajectories along which space objects move in gravitational fields. What specific type of section is the trajectory of the cosmic bodies of the system is determined by the ratio of their masses, absolute velocities and the distances between them.