A cone (more precisely, a circular cone) is a body formed by the rotation of a right triangle around one of its legs. Being a three-dimensional body, the cone is characterized, among other things, by volume.
This volume must be able to calculate.
The cone can be defined in different ways. For example, the radius of its base and the length of the side can be known. Another option is the radius of the base and the height. Finally, another way to define a circular cone is to specify the angle at its vertex and height. As is easily seen, all these methods determine the circular cone uniquely.
The base radius and height are most often knownThe cone. In this case, you first need to calculate the base area. According to the formula of the circle, it will be equal to? R ^ 2, where R is the radius of the base of the cone. Then the volume of the whole body is equal to? R ^ 2 * h / 3, where h is the height of the cone. This formula is easy to verify with the help of integral calculus. Thus, the volume of a circular cone is exactly three times smaller than the volume of a cylinder with the same base and height.
If the height is not specified, and instead knownThe radius of the base and the length of the side, then to determine the volume, you first have to find the height. Since the side is a hypotenuse of a right triangle, and the radius of the base is one of its legs, the height will be the second leg of the same triangle. By the Pythagorean theorem, h =? (L ^ 2 - R ^ 2), where l is the length of the lateral side of the cone. Obviously, this formula will only make sense if l? R. In this case, if l = R, then the altitude turns to zero, since the cone in this case turns into a circle. If l & lt - R, then the existence of such a cone is impossible.
If the angle at the vertex of the cone and itsHeight, then to calculate the volume you will need to find the radius of the base. To do this, it is necessary to turn to the geometric definition of the cone as the body formed by the rotation of a right triangle. In this case, the known angle at the vertex will be twice as large as the corresponding angle of this triangle. Therefore, the angle at the vertex is conveniently denoted by £. Then the angle of the triangle will be?.
By the definition of trigonometric functions,The required radius is l * sin (?), Where l is the length of the lateral side of the cone. At the same time, the height of the cone, known from the assumption of the problem, is l * cos (?). From these equalities it is easy to deduce that R = h / cos (?) * Sin (?) Or, which is the same, & nbsp-R = h * tg (?). This formula always makes sense, since the angle?, Being an acute angle of a right triangle, will always be less than 90 °.