In the stereometry course of secondary schools, several rotation figures in three-dimensional space are studied. One of them is a cone. This article discusses in detail the question of how to find the volume of a cone through the area of the base of the figure.
What is a cone?
Before giving the formula for the volume of the cone through the base area, we give a definition of this figure. In stereometry, a conical surface is understood to mean such a surface that the movement of a straight segment is fixed, fixed at one point at one point, along a known flat curve. By a flat curve, which is called a directrix, we mean any curve line lying in one plane. The segment that describes the surface is called the generatrix.
The conical surface itself does not have volume. However, if the director is a closed curve, then she defines a planar figure that bounds the conical surface from below. The combination of a conical surface and a figure bounded by a directrix (base) form a cone. He already has some volume in space.
The shape of the base may be arbitrary. The most common are the ellipse and the circle. The corresponding cones are called elliptical and circular. The first one is shown above.
General cone volume formula
How to find the volume of the cone? Very simple, it is enough to know only its height and the area of the base. Height is the length of the perpendicular that is lowered from the top of the figure to its base. The apex of the cone is the point bounding the conical surface from the side opposite the base.
Denote the height by h, and the base area by S o , then the volume of the cone through the base area is written as:
V = 1/3 * h * S o .
This formula is valid for absolutely any type of cone, including inclined figures.
Volumes of circular and elliptical cones
The circular cone is perhaps the most striking representative of this class of figures. It can be obtained by rotating a triangle with a right angle around any leg. The base of this cone is a circle, so its volume can be calculated as follows:
V = 1/3 * h * pi * r 2 .
Where r is the radius of the round base.
An elliptical cone also sometimes appears in geometric problems, so it is useful to give a volume formula for it. Remembering how the area of the ellipse is calculated, we obtain the following expression for the volume of the cone through the base area:
V = 1/3 * h * pi * a * b.
Here a, b are the lengths of the semiaxes of the ellipse.
Note that the general formula for the volume of the cone has a form similar to that for the pyramid.