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 that relate the linear parameters of a cone to 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 some point in space with all points of a smooth flat curve. This curve can be a circle or an ellipse. The figure below shows the cone.
The presented figure does not have volume, since the walls of its surface have an infinitely small thickness. However, if it is filled with matter and limited 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 called a cone.
The shape of the cone can often be found in life. So, it has an ice cream cone or striped black-and-orange traffic cones, which are put on the roadway to attract the attention of traffic participants.
Elements of the cone and its types
Since the cone is not a polyhedron, the number of elements forming it is not so large as for polyhedra. In geometry, a general cone consists of the following elements:
- bases whose bounding curve is called a directrix, or generatrix;
- a lateral surface, which is the set of all points of straight segments (generatrices) connecting the vertex and the points of the guide curve;
- vertex, which is the intersection point of the generatrix.
Note that the vertex in the base plane should not lie, since in this case the cone degenerates into a flat figure.
If we draw a perpendicular segment from the top to the base, then we get the height of the figure. If the last base intersects at the geometric center, then this 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 inclined 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 on this axis lies, and for an inclined figure, the height with the axis forms a certain angle. The axis of the cone is indicated by the letter a.
Straight cone with a round base
Perhaps this is the most common cone of the considered class of figures. It consists of a circle and a side surface. It is not difficult to obtain it by geometric methods. To do this, take a right-angled 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 shows the described scheme for obtaining the considered rotation pattern.
The depicted triangle can be rotated around another leg, which will result in a cone with a large base radius and lower height than the first.
To uniquely determine all the parameters of a round straight cone, one should know any two of its linear characteristics. Among them, the radius r, height h, or the length of the generator g is 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:
g 2 = r 2 + h 2 .
Surface area
When studying the surface of any three-dimensional figure, it is convenient to use its scan to the plane. The cone is no exception. For a round cone, a sweep is shown below.
We see that the sweep 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 the circle is easy to find, and the corresponding formula is known to every student. Speaking of the circular sector, we note that it is part of a circle with radius g (the length of the generatrix of the cone). The arc length of this sector is equal to the circumference of the base. These parameters allow you to uniquely determine its area. The corresponding formula is:
S = pi * r 2 + pi * r * g.
The first and second terms in the expression are the base cone 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 in the form:
S = pi * r 2 + pi * r * √ (r 2 + h 2 ).
Figure volume
If you take a straight pyramid and increase at infinity the number of sides of its base, 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 the pyramid when calculating a similar value for the cone. The volume of the cone can be found by the formula:
V = 1/3 * h * S o .
This formula is always true, regardless of what constitutes the base of the cone, having an area S o . Moreover, the formula also applies to an inclined cone.
Since we are studying the properties of a straight figure with a round base, we can use this expression to determine its volume:
V = 1/3 * h * pi * r 2 .
The validity of the formula is obvious.
The task of finding surface area and volume
Let a cone be given whose radius is 10 cm and the length of the generator is 20 cm. It is necessary to determine the volume and surface area for this figure.
To calculate the area S, you can immediately use the formula written above. We have:
S = pi * r 2 + pi * r * g = 942 cm 2 .
To determine the volume, you need to know the height h of the figure. We calculate it using the connection between the linear parameters of the cone. We get:
h = √ (g 2 - r 2 ) = √ (20 2 - 10 2 ) ≈ 17.32 cm.
Now you can use the formula for V:
V = 1/3 * h * pi * r 2 = 1/3 * 17.32 * 3.14 * 10 2 ≈ 1812.83 cm 3 .
Note that the volume of the round cone is one third of the cylinder into which it is inscribed.