  We establish the equivalence of the Tuynman midpoint area formula for a
spherical triangle to the classical area formulas of Euler and of Cagnoli. The
derivation also yields a variant of the Cagnoli formula in terms of the medial
triangle.
  We introduce the three barycentric coordinates of a point within the
spherical triangle as area ratios which sum to unity. The barycentric
coordinates are the Whitney 0-forms, scalar functions over the domain of the
triangle associated with each vertex. We then construct, by exterior
differentiation of the barycentric coordinates, succinct expressions for the
Whitney 1-forms associated with each geodesic side, or great circle arc.
  The Euler formula, in conjunction with that of Tuynman, facilitates the
differentiation of a triangular area with respect to the position of a vertex.
As both the Euler and Tuynman formulas may be expressed naturally in terms of
the position vectors of the vertices on an embedded sphere, the Whitney
constructions may be done in terms of vector-valued forms and without recourse
to a particular projection or coordinate chart.
  Finally, we exhibit the Whitney 2-form of the triangle, which must be the
product of a scalar function and the area 2-form of the sphere. We find an
expression for this scalar function in terms of determinants of $3 \times 3$
matrices built from the vertex position vectors. It is a rational function in
the Cartesian coordinates of a point. By construction it must be invariant
under cyclic permutations of the three vertices, though it is not manifestly
so. Also by construction the Whitney 2-form must integrate to one over the
triangle. Thus the rational function is exactly integrable over its triangle
when restricted to the sphere.
