The volume of the cone

Components of the cone

In order to know the volume of a cone, it is necessary to know what it consists of. The base of the geometric body and the vertex are the main generators of this geometric figure.

Straight lines connecting the vertex of the cone with the boundary of the base are called generators.

The forming (conical) or lateral surface of the cone is the union of all generators. The height of the figure is the straight line that connects the vertex and the base of the cone at a right angle to the base. The straight line that connects the top and center of the base is called the axis. You should also know that the angle between the two opposite components is called the angle of the solution.

Kinds

For a figure such as a cone, the volume of mathematics is calculated using different formulas, which vary depending on its type. When it comes to the cone, most imagine a circle at the base and a sharp top. But this is a delusion of people who have forgotten the course of the school curriculum. The shape of the cone, when its base forms a circle, is called circular. If the polygon lies at the base of the cone, then this will already be a pyramid. If there is an ellipse, a hyperbola or a parabola in the base, then such a figure is called an elliptic, hyperbolic, and parabolic cone, respectively. The last two cases have an infinite volume of a cone.

The varieties of this geometric figure can be divided into the following types: a regular and an incorrect cone. The second case assumes that the vertex with the geometric center of the base is connected by a straight line perpendicular to this base, which is a circle or a regular (equilateral) polygon. For example, a perpendicular line connects the center of a circle or the intersection of the diagonals of a square with a vertex. If the vertex is offset with respect to the symmetrical center of the base of this geometric figure, then it is denoted as oblique.

In addition, there is a truncated cone (truncated pyramid), which, based on the definition of the school geometry course, is not a separate geometric figure, but is only part of the whole cone (pyramid). In other words, a plane that is parallel to the plane of the base cuts off a smaller cone from the cone, and the remaining part is a truncated cone. However, another definition from the school program completely differently treats the concept of a truncated cone as a separate geometric figure (in the case of a circular one): a body formed by rotating a rectangular trapezium around the lateral side, which forms right angles with the trapezium bases.

The volume of the cone and the truncated cone

Greek scientists have long ago derived formulas that help accurately calculate the volume of both the cone and the truncated part of it.

In order to calculate the volume of the cone, we need to multiply the area of the base by the height of the cone, and then divide the obtained product by three. Private, which we will get, and will be the area of the cone. Exactly the same formula also serves to calculate the volume of a pyramid, as a special case of a cone. On paper, the formula is as follows: O = CXB / 3, where C is the base area, B is the height.

For a geometric figure "truncated cone," the volume is calculated by a more complex formula, which, however, is also not something beyond the bounds and complex. The sum of the radii of the bases, squared, is summed with the product of the radii of the bases. The resulting number is multiplied by a constant number π (3,14) and then multiplied by the height. The result of the product is divided by 3. The formula for calculating the volume will look like this on paper: O = BXrX (P1XP1 + P1XP2 + P2XP2) / 3. In this formula, B is the height of the truncated cone, P1 is the radius of the lower base, P2 is the radius of the upper base, and π is a constant number (3.14).