Application

As children, we all loved ice creams, especially the ice-creams which came in the waffle cones. Waffle pastry is an example of a cone. Here we will study about the shape of cones and learn various equations related to cones.

Definition

The definition of a cone is anything with a circular surface on one end and one point at the other end where all sides or lines meet.

A cone is equivalent to a right pyramid whose base is a circle. The lateral surface of a cone does not consist of triangles like in a right pyramid but is a single curved surface

Slant height

The slant height (or lateral height) of a right circular cone is the distance from any point on the circular base to the apex of the cone via a straight line along the surface of the cone.

If r is the radius of the base of the cone, h is the height of the cone and l is the slant height of the cone, then we have the relationship.

L2 = r2 + h2

This equation can be proved using the Pythagoras theorem.

Volume

If r is the radius of the base of the cone, h is the height of the cone, then it’s base area will be given by , and consequently it’s volume becomes

Volume = 1/3

The lateral surface area of a cone is the area of the curved or lateral surface only and is given by

Lateral surface area =

The total surface area is the sum of the lateral or curved surface area and the area of the base and is given by

Total surface area = curved surface area + base area

CONE FRUSTUM

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of the cone.

A very good example of a cone frustum are the buckets we use at our homes to store water.

If r is the top radius, R is the bottom or base radius, h is the height between the cut portion and the base and l is the slant height, then we have,

Lateral surface area of the cone =

Total surface area of cone = 2 + r2 + R.l + r.l)

Volume = 1/3 * 2 + R.r + r2)

And the equation of slant height becomes l2 = (R – r )2+ h 2