Math CentralQuandaries & Queries

Question from Aparna:

The central angles of two sector of a circle are 60° and 120° . Using these sectors two cones are made. If the radius of the smaller cone is 5 cm, find the radius and base area of the larger cone.


Look at the diagrams in my earlier response to Tasha. In these diagrams the green sector of the circle on the left is used to form the cone on the right. Notice that the arc $AB$ of the sector forms the circumference of the circular base of the cone on the right. Use the formula for the length of an arc in terms of the central angle to show that if the measure of the central angle is doubled then the length of the arc is also doubled. Thus the lengths of the circumferences of the base circles of the two cones in your problem have a ratio of 2:1. Show that the radii of these circles also have a ratio of 2:1.


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