2019 MPFG Problem 17: Difference between revisions
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Here is a demonstration of the actual transformation | Here is a demonstration of the actual transformation | ||
[[File | [[File:2019MPFG 17.jpg|450px|center]] | ||
As we can see, the transformation creates a rectangular prism with <imath>4</imath> triangular pyramids cut off from the corners. | As we can see, the transformation creates a rectangular prism with <imath>4</imath> triangular pyramids cut off from the corners. | ||
Latest revision as of 08:55, 7 November 2025
Problem
Let 
be a right prism whose two bases are equilateral triangles with side length 
. The height of is 
. Let 
be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases 
about . Let 
be the convex hull of the two current triangles. What is the volume of ?
Solution 1
Here is a demonstration of the actual transformation
As we can see, the transformation creates a rectangular prism with 
triangular pyramids cut off from the corners.
The volume of the rectangular prism is
![\[2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12\]](http://latex.artofproblemsolving.com/7/2/9/729378ba76c799ca473156217c6cec824426c415.png)
Subtract the volume of the triangular pyramids, and we get:
![\[V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}\]](http://latex.artofproblemsolving.com/d/0/9/d094512789de79b27e7ed2695578d2f9b3657530.png)
![\[= 12 - 8 = \boxed{4}\]](http://latex.artofproblemsolving.com/4/0/3/403f643c2e93b50c95b4fc663d020315a913cab0.png)
~cassphe
