Bung that fits Triangular, Square and Circular Holes

Problem
What shape of bung can be used to plug three different holes:
 * one square
 * one triangular
 * one circular?

Solution
Let $r$ be the radius of the circle we want to plug.

Take a cylindrical piece $B$ of the material for your bung whose base has radius $r$.

This will plug the circular hole.

Cut $B$ so that its height is $2 r$.

When you hold $B$ sideways, you see it will plug a square hole of side $2 r$.

Now draw a diameter $D$ of one of the bases of $B$.

Take two planes through $B$, both of which pass through $D$ such that they are both tangent to the other bases.

Slice away the material of $B$ on the other sides of the planes from the center of $B$.

Now $B$ has a triangular cross-section and it will plug a triangle whose sides are $\tuple {2 r, \sqrt 5 r, \sqrt 5 r}$.


 * Multi-Bung.png