Polygon Prism Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A lead pencil whose ends are regular hexagons was cut from a cylindrical piece of wood, with the least waste. If the original piece was 8 in. long and ½ in. in diameter, find the volume of the pencil.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The first thing that we need to do is to consider the base of the pencil which is a regular hexagon and draw three diagonals that passes thru its center as follows

Photo by Math Principles in Everyday Life

Next, we need to consider a triangle in a regular hexagon so that we can solve for its area as follows

Photo by Math Principles in Everyday Life

Since all the interior angles of a triangle are congruent, then it is an equilateral or equiangular triangle. All sides of an equilateral triangle are congruent. The base of a triangle is also equal to ¼ in.

Photo by Math Principles in Everyday Life

The area of a triangle is

Hence, the area of the base of a regular hexagon is

Therefore, the volume of a pencil which is a regular hexagon prism is  

or

Square Prism Problems

Category: Solid Geometry

"Published in Newark, California, USA"

Write a formula for (a) the volume, (b) the total area of a right prism whose altitude is h and whose base is a square of edge a.

Solution:

To illustrate the problem, it is better to draw the figure as follow

Photo by Math Principles in Everyday Life

(a) The volume of a square prism is

 

 The area of the base of a square prism is

The lateral area of a square prism is
 

 

(b) The total area of a square prism is

 

 

Triangular Prism Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A kite is in the form of an equilateral triangle of edge 3 ft. During one part of its flight, its center moves on a straight line while the kite remains parallel to its first position. If the distance between the planes of the first and last positions of the kite is 100 ft., find the volume swept out by the kite during this part of its flight.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Consider the base of a triangular prism which is the kite as follows

Photo by Math Principles in Everyday Life

The area of the base which is the kite is

Therefore, the volume of a triangular prism which is the volume swept out by the kite during its flight is

or