SOLUTION: A circular sheet of paper of radius 6 cm is cut into three equal sectors, and each sector is formed into a cone with no overlap. What is the height in centimeters of each cone?

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Question 98505: A circular sheet of paper of radius 6 cm is cut into three equal sectors, and each sector is formed into a cone with no overlap. What is the height in centimeters of each cone?
Answer by Earlsdon(6294) (Show Source):
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We could find the height of the resulting cone if we knew their slant-height and radius.
Starting with the circular sheet of paper whose radius (R) is 6 cm. If you divide this into three equal sectors to form the cones, then each cone will have a slant-height equal to the radius of the original circle, right?
Now the circumference of the base of each cone will be equal to one third of the circumference of the original circle.
The circumference (C) of the original circle is:
and, of course, you know that R is 6 cm, so...
Divide both sides by 3 to find one third of this.
and this is the circumference (c) of the cone base.
From this we can find the radius (r) of the cone base because but this is equal to . So we can write:
Simplifying, we get:
and so...
this is the radius of the cone base.
Now we can use the Pythagorean theorem to find the cone height using the slant-height of the cone (6 cm) as the hypotenuse of a right triangle and the radius of the cone base (2 cm) as the base of a right triangle.
where h is the height of the cone.
Subtract 4 from both sides.
Take the square root of both sides.
cm. This is the height of the cone to the nearest thousandth.