A pencil whose ends are regular hexagons was cut from a cylindrical piece of wood with the least waste possible. If the original piece was 8in long and 12in in diameter whats the volume of the pencil?

OK this is pretty easy even though it sounds hard.

Step 1. Determine the radius of the hexagon.

* If the cylinder of the pencil was 12 in diameter and the pencil was cut with the least waste, then the corners of the hexagon would touch the original outer surface of the cylinder. Simply put the diameter of the hexagon would be the same as the original cylinder. * If the diameter was 12 in then the radius is 6 in of course! Step 2. Determine the area of the hexagon.

* Short way: We can use the formula Area = 2.598 * radius2 (i.e 2.598 * 36)

* Long way: This way we learn something!! Assuming we are working this out on paper, draw a line from each corner of the hexagon through the center point to the exact opposite corner. * You will notice that you now have 6 equilateral triangles. Now we have two ways of continuing

We can find the area of one of those triangles using the formula √3 / 4 * (s * s) where s= length of one side. Or we can use another method which is cooler! * draw a line from the center point through the middle of one of the triangles to the mid point of one of the sides. This divides the equilateral triangle in half and gives us two equal sized right triangles.

* now we should know that the angles of an equilateral triangle always equal 60 degrees. So now we have a right triangle with 3 angles 90, 60, 30 degrees.

* Now we put the Pythagorean theorem to work for us. We know the length of the hypotenuse (6in) already as it is one of the sides. And we know the length of one of the other sides (3in) as it ends at the mid point of one of the sides. * Now using the theorem which is usually written a2 + b2 = c2 we already know c=6 and a=3 we need to solve for b.

* 32 + b2 = 62 * 9 + b2 = 36 * b2 = 36 - 9 * b2 = 27

* b= √27 Now we can find the area of the smaller right triangle because we know all the value of b. The formula is (length x height) / 2 = Area in square units.

* (√27 * 3) /2 = 7.794 sq in

Now if we take the area of our triangle and multiply it by 12 we will know the area of our hexagon! * The Area of our hexagon is 93.53 sq in Step 3. find the volume of the pencil

* Now we multiply the area of our hexagon by the length of the pencil * ** 93.53 * 8 = Volume of our pencil The answer is 748.25 cubic inches! Thats a big pencil!!