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Cones, Cylinders, and Spheres Help

By — McGraw-Hill Professional
Updated on Oct 25, 2011

Cone vs. Cylinder

A cone has a circular or elliptical base and an apex point. The cone itself consists of the union of the following sets of points:

  • The circle or ellipse.
  • All points inside the circle or ellipse and that lie in its plane.
  • All line segments connecting the circle or ellipse (not including its interior) and the apex point.

The interior of the cone consists of the set of all points within the cone. The surface might or might not be included in the definition of the interior.

A cylinder has a circular or elliptical base, and a circular or elliptical top that is congruent to the base and that lies in a plane parallel to the base. The cylinder itself consists of the union of the following sets of points:

  • The base circle or ellipse.
  • All points inside the base circle or ellipse and that lie in its plane.
  • The top circle or ellipse.
  • All points inside the top circle or ellipse and that lie in its plane.
  • All line segments connecting corresponding points on the base circle or ellipse and top circle or ellipse (not including their interiors).

The interior of the cylinder consists of the set of all points within the cylinder. The surface might or might not be included in the definition of the interior.

These are general definitions, and they encompass a great variety of objects! Here, we’ll look only at cones and cylinders whose bases are circles.

The Right Circular Cone

A right circular cone has a base that is a circle, and an apex point on a line perpendicular to the base and passing through its center. This type of cone, when the height is about twice the diameter of the base, has a “dunce cap” shape. Dry sand, when poured into an enormous pile on a flat surface, acquires a shape that is roughly that of a right circular cone. An example of this type of object is shown in Fig. 10-12.

Geometry in Space THE RIGHT CIRCULAR CONE

Fig 10-12. A right circular cone.

Surface Area of Right Circular Cone

Imagine a right circular cone as shown in Fig. 10-12. Let P be the apex of the cone, and let Q be the center of the base. Let r be the radius of the base, let h be the height of the cone (the length of line segment PQ ), and let s be the slant height of the cone as measured from any point on the edge of the base to the apex P . The surface area S 1 of the cone, including the base, is given by either of the following formulas:

S 1 = π r 2 + π rs

S 1 = π r 2 + π r ( r 2 + h 2 ) 1/2

The surface area S 2 of the cone, not including the base, is called the lateral surface area and is given by either of the following:

S 2 = π rs

S 2 = π r ( r 2 + h 2 ) 1/2

Volume of Right Circular Cone

Imagine a right circular cone as defined above and in Fig. 10-12. The volume, V , of the interior of the figure is given by:

V = π r 2 h /3

The Slant Circular Cone

A slant circular cone has a base that is a circle, just as does the right circular cone. But the apex is such that a line, perpendicular to the plane containing the base and running from the apex through the plane containing the base, does not pass through the center of the base. Such a cone has a “blown-over” or “cantilevered” appearance (Fig. 10-13).

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