Cones and Cylinders Help

Introduction to Cones and Cylinders

Cones

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 cone itself might or might not be included in the definition of the interior.

Cylinders

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 cylinder itself might or might not be included in the definition of the interior.

These are general definitions, and they encompass a great variety of objects! In this chapter, 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 that lies on a line normal to the plane of the base and that passes through the center of the base (Fig. 8-6).

Fig. 8-6 . A right circular cone.

Surface Area Of Right Circular Cone

Imagine a right circular cone as shown in Fig. 8-6. 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. 8-6. The volume, V , of the interior of the figure is given by:

V = π r ² h /3

Surface Area Of Frustum Of Right Circular Cone

Imagine a right circular cone that is truncated (cut off) by a plane parallel to the base. This is called a frustum of the right circular cone. Let P be the center of the circle defined by the truncation, and let Q be the center of the base, as shown in Fig. 8-7. Suppose line segment PQ is perpendicular to the base. Let r ₁ be the radius of the top, let r ₂ be the radius of the base, let h be the height of the object (the length of line segment PQ ), and let s be the slant height. Then the surface area S ₁ of the object (including the base and the top) is given by either of the following formulas:

S ₁ = π( r ₁ + r ₂ )[ h ² + ( r ₂ − r ₁ ) ² ] ^1/2 + π( r ₁ ² + r ₂ ² )

S ₁ = π s ( r ₁ + r ₂ ) + π( r ₁ ² + r ₂ ² )

The surface area S ₂ of the object (not including the base or the top) is given by either of the following:

S ₂ = π( r ₁ + r ₂ )[ h ² + ( r ₂ − r ₁ ) ² ] ^1/2

S ₂ = π s ( r ₁ + r ₂ )

Fig. 8-7 . Frustum of a right circular cone.