Cone

one of the geometrical solids. A right circular cone is a solid occupying the space through which a right-angled triangle passes when swung clear around on one leg; that is to say, when it turns on its heel. The discussion which follows pertains to a cone of this sort, not to an oblique cone. The base of a cone is circular. The opposite point is called the apex. If a portion next the apex be cut off by a plane, the remainder is a truncated cone. The volume of a frustrum may be found by subtracting from the volume of the complete cone the volume of the portion that has been removed. The lateral surface of a cone equals the product of the circumference of its base by half its slant height. The solid content or volume of a cone equals one-third the product of its base by its altitude, not its slant height.

Every section of a cone passing through the apex is a triangle. By passing a cutting plane through a cone in various ways four different sections, called conic sections, may be formed. They are the circle, the ellipse, the parabola, and the hyperbola. If the cutting plane be parallel to the base, the section is a circle. If the cutting plane pass obliquely through the cone, not parallel to the base, the section is an ellipse. If parallel to the side of the cone, the section is a parabola. If perpendicular to the base, but not passing through the apex, the section is a hyperbola.

The circumference of the circle is, of course, everywhere equally distant from the center. Instead of a center, an ellipse has two points within, called foci. The sum of the distances from the foci to any point in the circumference is always the same. The farther the point is from one focus, the nearer it is to the other. In theory, the ends of the arc of a parabola will meet somewhere in space if sufficiently extended. Comets are supposed to move in parabolic paths, by virtue of which they come within our vision at regular intervals of time. The cables of a suspension bridge when loaded uniformly are said by engineers to swing in the arc of a parabola. Leaving out of consideration the resistance of the atmosphere, a cannon ball or a jet of water from the spigot of a barrel describes a portion of a parabola. In theory, the arc of the hyperbola is never completed. A projectile following a hyperbola would travel on and on into space and never return.