Question:
Three identical cones with base radius \( r \) are placed so each touches the other two. What is the radius of the circle drawn through their vertices?
Three identical cones with base radius \( r \) are placed so each touches the other two. What is the radius of the circle drawn through their vertices?
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The circumradius of a triangle formed above a base circle (cone tip positions) is greater than the base radius.
Updated On: Aug 6, 2025
- smaller than \( r \)
- equal to \( r \)
- larger than \( r \)
- depends on height of cones
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The Correct Option is C
Solution and Explanation
Each cone touches the other two at their bases — means they form an equilateral triangle at the base level.
Now we consider the vertices (tips) of cones — these lie above the base.
When a circle is drawn through the vertices of three upright cones arranged in such a triangle, the circle will pass above the center and surround the triangle. Due to the geometry, the radius of this circle will depend on the spatial arrangement, but crucially, it must be larger than the base radius \( r \) to pass through all three cone vertices. \[ \boxed{\text{Larger than } r} \]
Now we consider the vertices (tips) of cones — these lie above the base.
When a circle is drawn through the vertices of three upright cones arranged in such a triangle, the circle will pass above the center and surround the triangle. Due to the geometry, the radius of this circle will depend on the spatial arrangement, but crucially, it must be larger than the base radius \( r \) to pass through all three cone vertices. \[ \boxed{\text{Larger than } r} \]
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