Cone
The surface generated by a moving straight line (generator) which always passes through a fixed point (vertex) and always intersects a fixed plane curve (directrix) is called conical surface. Cone is a solid bounded by a conical surface whose directrix is a closed curve, and a plane which cuts all the elements. The conical surface is the lateral area of the cone and the plane which cuts all the elements is the base of the cone.

 

Like pyramids, cones are generally classified according to their bases.
 

Right cone and Oblique Cone with Some Elements

 

Volume of a Cone

$V = \frac{1}{3}A_b \, h$

 

Properties of a Cone

  • An element of a cone is the generator in any particular position.
  • The altitude of the cone is the perpendicular drop from vertex to the plane of the base. It is denoted as h.
  • Every section of a cone made by a plane passing through its vertex and containing two points of the base is a triangle. See section PQV, where V is the vertex and P and Q are two points on the base.
  • The axis of the cone is the straight line joining the vertex with the centroid of the base. For right cone, altitude and axis are equal in length.
  • The right section of a cone is a section perpendicular to its axis and cutting all the elements. For right cone, the right section is parallel and similar to the base. Right section is denoted by AR.
  • A circular cone is cone whose right section is a circle.

 

The Right Circular Cone

Any cone with circular right section is a circular cone. Right circular cone is a circular cone whose axis is perpendicular to its base.
 

Right Circular Cone with Unrolled Lateral Area

 

Properties of Right Circular Cone

  • The slant height of a right circular cone is the length of an element. Both the slant height and the element are denoted by L.
  • The altitude of a right circular is the perpendicular drop from vertex to the center of the base. It coincides with the axis of the right circular cone and it is denoted by h.
  • If a right triangle is being revolved about one of its legs (taking one leg as the axis of revolution), the solid thus formed is a right circular cone. The surface generated by the hypotenuse of the triangle is the lateral area of the right circular cone and the area of the base of the cone is the surface generated by the leg which is not the axis of rotation.
  • All elements of a right circular cone are equal.
  • Any section parallel to the base is a circle whose center is on the axis of the cone.
  • A section of a right circular cone which contains the vertex and two points of the base is an isosceles triangle.

 

Formulas for Right Circular Cone

Area of the base, Ab
The bases of a right circular cone are obviously circles
$A_b = \pi r^2$

 

Lateral Area, AL
The lateral area of a right circular cone is equal to one-half the product of the circumference of the base c and the slant height L.
 

$A_L = \frac{1}{2}cL$
 

Taking c = 2πr, the formula for lateral area of right circular cone will be more convenient in the form

$A_L = \pi rL$

 

The relationship between base radius r, altitude h, and slant height L is given by

$r^2 + h^2 = L^2$

 

Volume, V
The volume of the right circular cone is equal to one-third the product of the base area and the altitude.
 

$V = \frac{1}{3}A_b h$
 

$V = \frac{1}{3} \pi r^2 h$

 

Derivation of Formula for Lateral Area of the Right Circular Cone

 
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