## Sunday, December 2, 2012

### Frustum - Right Circular Cone

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

The frustum of a right circular cone has a slant height of 9 ft. , and the radii of the bases are 5 ft. and 7 ft. Find the lateral area and the total area. What is the altitude of this frustum? Find the altitude of the cone that was remove to leave this frustum. Find the volume of the frustum.

Solution:

To illustrate the problem, you can draw the figure as follows Photo by Math Principles in Everyday Life

Since the slant height and the radii of the bases are given, we can get the lateral area, area of the bases, and the total area of the frustum.

Let's get the circumference of the top base of the frustum as follows

For the bottom base of the frustum

Therefore, the lateral area of the frustum is

Let's get the area of the top base of the frustum as follows

For the bottom base of the frustum

Therefore, the total area of the frustum is

Since the altitude of the frustum is not given in the problem, we have to find the altitude of the frustum with the use of the  vertical section of the frustum. The vertical section of the frustum is a trapezoid. Photo by Math Principles in Everyday Life

From the figure, we can get the altitude of a trapezoid which is also the altitude of a frustum. As you notice that the end portion of a trapezoid is a right triangle. We can use the Pythagorean Theorem to solve for the altitude of the frustum as follows

In order to get the altitude of the cone that was removed to leave this frustum, we have use the formula for similar triangles as follows

Finally, we can get the volume of the frustum as follows