The Smallest Equilateral Cover for Triangles of Perimeter Two

Abstract

In an obscure but interesting pamphlet [7], Josiah Smith' announced his belief every triangle of perimeter two can be covered by an equilateral triangle of side one. Experiments suggest, he wrote, that every triangle with perimeter two can be placed in an equilateral triangle of side one, although I cannot establish this fact in fullsome rigour. ... No smaller equilateral triangle has this property, because flat isosceles triangles of base 1 - 8 and equal sides 1 + 8 must fit. In this note we solve Smith's problem by determining the side of the smallest equilateral triangle (i.e., closed equilateral-triangular region) can cover eveiy triangle of perimeter two, and we discover Smith's intuition was not correct: a side longer than one is required. We show the smallest equilateral triangle To can cover every triangle with perimeter two has side so:= 2/rno, where mi0 is the global minimum of the trigonometric function