### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Let the large circles have radius R. $$Area A = 2 x \pi x ({1 \over 2}R)^2 = 1/2 \pi R^2$$ $$Area B = 3 x {1 \over 2} x R^2 x {\sqrt 3\over 2} = {3\over 4} \sqrt 3 R^2$$ $$Area C = (2 R^2) - \pi R^2 = (4 - \pi)R^2$$ $$Area D = 2 x {1 \over 2} x R^2 = R^2$$ $$Area E = \pi R^2 (\sqrt 2 R)^2 = (\pi - 2)R^2$$