A cone is an three-dimensional geometric shape that formed by the locus of all straight line segments that join the apex to the base. The surface generated by a straight line, the generator, passing through a fixed point and moving along a fixed curve. In elementary geometry, cones are assumed to be right circular. The volume and the surface area of the cone depend upon the area of base, height and slant height of the cone.

Cone Shape:

Cone

Where, 'r' be the radius, 'h' be the height and 'l' be the slant height of a cone.

The formulas for the surface area and volume of a cone are as follow:

Formula:

Curved Surface Area of Cone = $\pi$rl

Total Surface Area of a Cone = $\pi$ r(r + l)

Volume of a Cone = $\frac{1}{3}$$\pi r^2$h

Where,
r = Radius of a cone
h = Height of a cone
l = Slant height of a cone.

Frustum of a cone is the part of the cone, if a cone is cut by a plane parallel to its base, the portion of a solid between this plane and the base is known as frustum of a cone, which is shaped like a truncated cone. A frustum may be formed from a cone with a circular base by cutting off the tip of the cone with a cut perpendicular to the height, forming a lower base and an upper base that are circular and parallel.
Frustum Cone
In above figure, ABDE is a frustum of the cone.

Let us see some shapes which are similar to cone:

Ice-cream Cone:


Ice-Cream Cone

Funnel:


Funnel


Water Paper Cup:

Water Paper Cup
Cone is a pyramid with a circular cross section. A right cone is a cone with its vertex above the center of its base. A right cone of height and base radius oriented along the z-axis,
with vertex pointing up, and with the base located at z = 0. The equation of a cone that will open along the z-axis.

Cone along Z-axis
A cone that opens up along the z-axis will have the equation,

$\frac{x^2}{a^2}$ + $\frac{y^2}{b^2}$ = $\frac{z^2}{c^2}$

We can also find the equation of a cone that opens along one of the other axes all we need to do is make a slight modification of the equation.

A cone that opens up along the x-axis will have the equation,

$\frac{y^2}{b^2}$ + $\frac{z^2}{c^2}$ = $\frac{x^2}{a^2}$

A cone that opens up along the y-axis will have the equation,

$\frac{x^2}{a^2}$ + $\frac{z^2}{c^2}$ = $\frac{y^2}{b^2}$