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All about Right Circular Cones

A right circular cone is a three-dimensional figure with a circular base. The surface of the cone is generated by a line (the generatrix) passing through a fixed point on the circumference of the base and moving along a plane that does not intersect the base. If the plane intersects the circle, then we obtain an oblique circular cone. The intersection of the plane with the lateral surface of the cone is called the directrix of the cone. The point where the generatrix intersects directrix is called vertex of the cone. The slant height is the perpendicular distance between directrix and generatrix at any point on their lines of intersection with the lateral surface. The altitude is perpendicular distance between directrix and plane of base.

 

If we wish to generate a right circular cone, then we must have a plane which does not intersect the circumference of its base, as shown in Figure 1. The line joining any point on the circumference to any point on this plane will be part of the surface of the required cone; such a line is called a generatrix of the cone. If we take any other plane through V, it will cut out another portion of this surface; all points common to this plane and surface of solid lie on two circles, one in plane and one on surface, as shown at (a) in Figure 2. But if we take still another plane through V, say at angle θ to first plane as shown at (b), this will again cut out some other portion from surface; common points now lie on two circles but also on two ellipses as can be seen from Figure 3.

 

The slant height is defined as length perpendicular from apex to any directrix curve (plane cutting elliptical or hyperbolic cross-section). As shown in Figure 4 for an elliptical cross-section, if PQ is perpendicular from apex then we have l^2 = h^2 + b^2, where h is altitude and b is semi-minor axis length. For hyperbolic cross-sections there are no real foci so l^2 = h^2 - b^2 where b is now semi-major axis length. Unlike an ellipse or circle, there may be more than one value for slant height corresponding to given values for h and b (and hence r1 and r2).

 

In conclusion, remember that a right circular cone has a circular base while an oblique circular cone has a noncircular base. If you're ever stuck trying to remember which is which, just think "right" vs "oblique"—the word "right" has five letters, just like "circular," while "oblique" has seven letters, just like "noncircular." Now you'll never mix them up again!


FAQ

What are the properties of right circular cone?

Right circular cones have a number of interesting properties. They are rotationally symmetric about their axis, which means that they look the same no matter how you rotate them. They also have a circular base, which means that their cross-sections are always circles. Finally, they have a point at their apex, which is the point that is farthest away from their base.

How is right circular cone formed?

A right circular cone is formed when a plane intersects a cylinder in such a way that the intersection is a circle. The circle can be either the top or bottom of the cylinder, and the plane can intersect the cylinder at any angle.

What are the applications of right circular cone?

Right circular cones have a number of applications in mathematics and engineering. In mathematics, they can be used to represent three-dimensional figures in two dimensions. In engineering, they are often used in the design of towers and other structures.

How many faces does a right circular cone have?

A right circular cone has three faces: the base, the side, and the apex.

What is the difference between right circular cone and cone?

A right circular cone is a cone in which the base is a circle. A cone can have any shape for its base, but a right circular cone will always have a circular base.

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