Math CentralQuandaries & Queries

Question from Jessica:

A circle has a radius of 7.5cm. A sector with an angle of 240 degrees is cut out from the sector. If the sector is folded to form a cone. Find the length of the cone.

Hi Jessica,

In the circle below of radius 7.5 cm I have cut out a sector with center angle 240 degrees from which I want to construct a cone.


Since $\large \frac{240}{360} \normalsize = \frac{2}{3}$ the symmetry of the circle tells us that the length of the arc forming the green section is two-thirds of the circumference of the circle.

The cone below is formed from the green sector of the circle above.


$|AB|,$ sometimes called the slant height of the cone is the radius of the circle above, 7.5 cm. The circle tat forms the base of the cone has a circumference which is the arc formed by the green sector in the first diagram. Since you know the length of this arc and an expression for the circumference of a circle you can calculate the radius, $|BC|$ of the base of the cone.

Finally triangle $ABC$ is a right triangle and you know the lengths of two of its sides you can use Pythagoras Theorem to find the length of the third side, the length of the cone.


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