Cone and Frustrum of a Cone

What is a cone?

A cone is a solid that has a circular base and a single vertex.

Types of Cones

Right Cone-  Vertex is over the center of the base.

Oblique Cone-  Vertex is not over the center of the base

Volume of Cone

The volume of a cone is given by the formula:

Where: r is the radius of the circular base,  h is the height - the perpendicular

distance from the base to the vertex.

Same for both right angle and oblique.

Volume =

Example 1:

Example 2:

The diagram below shows a conical container filled with water. The base of the cone lies on a horizontal table. The volume of water is 821 1/3 cm3.

Using pi = 22/7, calculate the height, in cm, of the water in the container.

Surface Area of Cone

The surface area of a cone is given by the formula

Where:r is the radius of the circular base,  s is the slant height of the cone.

But there are no formulas for oblique cone.

Frustum of Cone

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base).

To find the slant height, we have to use

The formula of the surface area isArea=

without the area for both circles.

Frustum of Cone

The formula for frustum of a pyramid or frustum of a cone is given by

Where:h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum)A1 = area of the lower baseA2 = area of the upper baseNote that A1 and A2 are parallel to each other.

Derivation

http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum

Summary:

Calculate the: 1) Volume of the cone. 2) Surface area of the cone. 3) Volume of the conical frustrum. 4) Surface area of the conical frustrum.

Height of water = 7