A cone is one of the spatial figures of rotation, the characteristics and properties of which are studied by stereometry. In this article, we will define this figure and consider the basic formulas connecting the linear parameters of a cone with its surface area and volume.
What is a cone?
From the point of view of geometry, we are talking about a spatial figure, which is formed by a set of straight segments connecting a certain point in space with all points of a smooth flat curve. This curve can be a circle or an ellipse. The figure below shows a cone.
The presented figure has no volume, since the walls of its surface have an infinitesimal thickness. However, if it is filled with a substance and bounded from above not by a curve, but by a flat figure, for example, a circle, then we will get a solid volumetric body, which is also commonly called a cone.
The shape of a cone can often be found in life. So, it has an ice cream cone or striped black and orange traffic cones that are put on the roadway to attract the attention of traffic participants.
Elements of a cone and its types
Since the cone is not a polyhedron, the number of elements forming it is not as large as for polyhedra. In geometry, a general cone consists of the following elements:
- base, the bounding curve of which is called the directrix, or generatrix;
- side surface, which is a collection of all points of straight line segments (generatrices) connecting the vertex and points of the guide curve;
- vertex, which is the intersection point of the generatrices.
Note that the vertex must not lie in the plane of the base, since in this case the cone degenerates into a flat figure.
If we draw a perpendicular segment from the top to the base, we will get the height of the figure. If the last base intersects at the geometric center, then it is a straight cone. If the perpendicular does not coincide with the geometric center of the base, then the figure will be inclined.
Straight and oblique cones are shown in the figure. Here, the height and radius of the base of the cone are denoted by h and r, respectively. The line that connects the top of the figure and the geometric center of the base is the axis of the cone. It can be seen from the figure that for a straight figure, the height lies on this axis, and for an inclined figure, the height forms an angle with the axis. The axis of the cone is indicated by the letter a.
Straight cone with round base
Perhaps, this cone is the most common of the considered class of figures. It consists of a circle and a sidesurfaces. It is not difficult to obtain it by geometric methods. To do this, take a right triangle and rotate it around an axis coinciding with one of the legs. Obviously, this leg will become the height of the figure, and the length of the second leg of the triangle forms the radius of the base of the cone. The diagram below demonstrates the described scheme for obtaining the rotation figure in question.
The depicted triangle can be rotated around another leg, which will result in a cone with a larger base radius and a lower height than the first one.
To unambiguously determine all parameters of a round straight cone, one should know any two of its linear characteristics. Among them, the radius r, the height h or the length of the generatrix g are distinguished. All these quantities are the lengths of the sides of the considered right-angled triangle, therefore, the Pythagorean theorem is valid for their connection:
g2=r2+ h2.
Surface area
When studying the surface of any three-dimensional figure, it is convenient to use its development onto a plane. The cone is no exception. For a round cone, the development is shown below.
We see that the unfolding of the figure consists of two parts:
- The circle that forms the base of the cone.
- The sector of the circle, which is the conical surface of the figure.
The area of a circle is easy to find, and the corresponding formula is known to every student. Speaking about the circular sector, we note that itis part of a circle with radius g (the length of the generatrix of the cone). The length of the arc of this sector is equal to the circumference of the base. These parameters make it possible to unambiguously determine its area. The corresponding formula is:
S=pir2+ pirg.
The first and second terms in the expression are the cone of the base and the side surface of the area, respectively.
If the length of the generator g is unknown, but the height h of the figure is given, then the formula can be rewritten as:
S=pir2+ pir√(r2+ h2).
The volume of the figure
If we take a straight pyramid and increase the number of sides of its base at infinity, then the shape of the base will tend to a circle, and the side surface of the pyramid will approach the conical surface. These considerations allow us to use the formula for the volume of a pyramid when calculating a similar value for a cone. The volume of a cone can be found using the formula:
V=1/3hSo.
This formula is always true, regardless of what the base of the cone is, having area So. Moreover, the formula also applies to the oblique cone.
Since we are studying the properties of a straight figure with a round base, we can use the following expression to determine its volume:
V=1/3hpir2.
The formula is obvious.
The problem of finding the surface area and volume
Let a cone be given, the radius of which is 10 cm, and the length of the generatrix is 20see Need to determine volume and surface area for this shape.
To calculate the area S, you can immediately use the formula written above. We have:
S=pir2+ pirg=942 cm2.
To determine the volume, you need to know the height h of the figure. We calculate it using the relationship between the linear parameters of the cone. We get:
h=√(g2- r2)=√(202- 102) ≈ 17, 32 cm.
Now you can use the formula for V:
V=1/3hpir2=1/317, 323, 14102 ≈ 1812, 83cm3.
Note that the volume of a round cone is one third of the cylinder it is inscribed in.
