All spatial figures studied by a special section of geometry - stereometry, can be divided into two types according to their shape: polyhedra (polyhedra) and figures formed by smooth surfaces. One of the figures of the second type will be discussed in the article. We popularly answer the question "what is a cone."
Figure definition
Suppose that there is some plane closed curve, for example a circle or an ellipse. In space, we choose some point that does not belong to the plane of the specified curve. Now connect it to each point on the curve using straight lines. The resulting figure gives an answer to the question of what a cone is. The photo below shows three cones made of paper.
The original closed curve is called the directrix or guide figure. The fixed point in space mentioned above is called the vertex of the cone. The straight segments that connect the vertex to the points on the directrix are called generators, or generators.
The figure in question forms a certain surface; therefore, it does not have a volume. If inside this figure is filled with some substance, then it will have some volume. The resulting solid is also called a cone.
Cone elements
Under the elements of the figure understand the geometric objects of which it consists. Knowing what a cone is, we can say that its main elements are the following:
- The base is a flat figure bounded by a director. For example, it could be a circle.
- Side surface. It is also called conical and is formed by the set of points of all generatrices.
- A vertex is a point that does not belong to the base at which all generatrices intersect.
A cone differs from polyhedra (polyhedra) in that it has no faces and edges, since it is formed by a smooth conical lateral surface. From a mathematical point of view, a cone is a pyramid with an infinite number of sides.
Types of cone
We learned what a cone is. Now let's move on to the question of what kinds of figures are.
Depending on the curve at the base, they talk about a round cone, elliptical, hyperbolic, parabolic and so on. In addition, the figure can be straight and inclined. To understand the difference between them, you should get acquainted with the concept of height.
The height of the cone is the length of the segment, which is perpendicular from the top of the figure is lowered to the plane of its base. If the height crosses the base at the geometric center (the center of a circle or ellipse), then the cone is called straight. Otherwise, they talk about a sloping figure. The figure below shows two cones. The left one is straight, the right one is sloping.
In most geometric problems, a straight cone with a round base is considered. Further in the article we give a detailed description of this figure.
How can I get a round straight cone?
The cone was defined above and one of the methods of its geometric construction. Since a round straight cone is a body of revolution, it can be obtained in the following simple way: we take an arbitrary triangle with a right angle. Suppose that the two legs of this triangle are equal to a and b. We put a triangle on one of the legs, for example a. And we will rotate it around the second leg b. The hypotenuse of the triangle in this case will describe the conical surface.
As a result of the described method for producing a round cone, a figure is formed having a radius of the base a and a height b. The leg b is part of the axis of the cone, which passes through its top and center of the base. The hypotenuse of the original triangle will be the generator of the figure.
The diagram above shows how you can get a cone by rotating a right triangle around one of the legs.
Linear characteristics of a round straight cone
The figure is formed by a circle of some radius r and a conical surface. Let the height of the cone be equal to h. These two linear characteristics are basic. Their knowledge allows you to calculate any parameters of the figure, for example, the length of its generatrix, surface area and volume.
Since the figure in question is straight, the lengths of all its generatrices are equal to each other. If we denote their length by the letter d, then the formula for its calculation will look like:
d = √ (h 2 + r 2 ).
It is easy to guess where this formula came from. It is the result of applying the Pythagorean theorem to the corresponding right-angled triangle. Note that the generatrix of a cone is always larger than the radius of its base, regardless of the value of h.
This expression allows us to determine the third from two known linear quantities. For example, if d and h are known, then the radius of the circle at the base will be:
r = √ (d 2 - h 2 ).
Surface and volume
As noted, the surface area of the figure is equal to the sum of the areas of its base and side surface. The formula for the base area is the same as for the circle. As for the side surface, if you present it in the form of a scan, you can see what it represents as a sector of a circle of radius d. Here d is the length of the generator. A cone scan is shown below.
Let the height of the cone be equal to h, and its radius of the base be r, then the following equality holds for the surface area of the figure in question:
S = pi * r 2 + pi * r * √ (r 2 + h 2 ).
Here, the first term reflects the area of the base, the second term - the area of the conical surface. You can see that the radical expression corresponds to the length of the generator d. The formula for the area of the conical surface is obtained if we consider the parameters of the circular sector shown in the scan above. Note that the arc length of this sector is equal to the directrix of the base of the figure.
The volume of a cone of arbitrary type is calculated by the following formula:
V = 1/3 * h * S o .
Here, the symbol S o denotes the area of the base. Note that a similar formula also applies to the volume of the pyramid. This coincidence is not accidental, since an increase in the number of faces of the pyramid to infinity translates it into a cone.
The written formula for the case of a straight round cone takes on a specific form:
V = pi / 3 * h * r 2 .
Here the factor pi * r 2 is the area of the base (circle).
Thus, the volume of the straight cone, the base of which is a circle, is equal to one third of the volume of the cylinder having the same radius and the same height.