Geometry: Right Circular Cone

A right circular cone is a solid generated by revolving a line segment which passes through a fixed point and which makes a constant angle with a fixed line.

Height of the Cone: The length of the segment VO is called the height of the cone and is denoted by h.

Slant Height of the Cone: The length of the segment VA is called the slant height of the cone and is denoted by l.

Radius of the Cone: The radius OA of the base circle is called the radius of the cone and is denoted by r.
Therefore,
l = (h2 + r2)

Surface Area
Curved Surface Area of Cone = rl
Total Surface Area of Cone = Curved Surface Area + Area of the base =r(l + r)
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Example: If h, C, V are respectively the height, the curved surface and the volume of a cone,prove that 3Vh3 - C2h2 + 9Vh2 = 0.
Solution: Let r and l denote respectively the radius of the base and slant height of the cone. Then,
l = (h2 + r2), V = 1/3r2h and C = rl

 3Vh3 - C2h2 + 9Vh2 = 3 x 1/3r2h x h3 - (rl)2h2 + 9 x (1/3r2h)2 = 2r2h4 - 2r2l2h2 + 2r4h2 = 2r2h4 - 2r2h2(r2 + h2) + 2r4h2 = 2r2h4 - 2r4h2 - 2r2h4 + 2r4h2 = 0.

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