"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
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| 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
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| 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
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| 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.
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| 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
