3420.24 – Three Equilateral Triangles

Suppose that two equilateral triangles have side lengths of xx and yy. Find the side length of a third equilateral triangle whose area is the sum of the areas of the first two triangles.


The area of an equilateral triangle of side ss is s23/4s^2 \sqrt{3}/4. Thus the areas of the first two triangles are x23/4x^2 \sqrt{3}/4 and y23/4y^2 \sqrt{3}/4. The sum of these is

(x2+y2)34,\frac{(x^2 + y^2) \sqrt{3}}{4},

so each side of the third triangle will be x2+y2\sqrt{x^2 + y^2}.