Another alternative route to calculating the answer. This uses 3 pentagon formulas found at
http://mathworld.wolfram.com/Pentagon.html.
A (area) = (1/4) * (a^2) * sqrt(25 + 10 * sqrt(5))
R (circumradius) = (1/10) * a * sqrt(50 + 10 * sqrt(5))
r (inradius) = (1/10) * a * sqrt(25 + 10 * sqrt(5))
where a is the length of a side of the pentagon.
Substituting 800 for A in the first equation, we get a = 21.563562. Using this value, we get R = 18.343062 and r = 14.839849.
Now, my wife pointed out the key fact to me last night, which is that if you reflect the green triangle inward, then the base is still one side of the pentagon, and the opposite vertex touches the opposite vertex of the pentagon. Then the area of the triangle is (1/2) * a * (r + R).
Both the conclusion about the geometry and the formula for calculating the area of the triangle need a bit of justification.
To show that the vertex of the reflected triangle is coincident with the vertex of the pentagon, we look at the angles involved. The interior angles of the pentagon are 108 degrees. Then the base angles of the triangle are 180 - 108 = 72 degrees (sum of angles making up a straight line is 180 degrees) and the top angle of the triangle is 180 - 72 - 72 = 36 degrees (sum of angles of a triangle is 180 degrees).
To allow talking about it, I'm going to name the edges of the pentagon as a, b, c, d, and e, starting with the edge shared with the green triangle and proceeding clockwise. I'm going to name the side of the triangle that is connected at the vertex shared by a and b as s.
When we reflect the triangle, the base angle is still 72 degrees and the angle between the side of the triangle and the adjacent edge of the pentagon is 108 - 72 = 36 degrees. (This is the angle between b and the reflected s, or s'). Now, b along with the extensions of c and s', forms a triangle, call it T. (So far, we don't assume that the vertices of the triangle and pentagon match.) Call the edges of this new triangle b, c" and s". Now the angle between b and s" = 36 degrees (same as the angle between b and s'), while the angle between b and c" is 108 degrees (same as the angle between b and c - an interior angle of the pentagon), so the third angle of T must be 180 - 108 - 36 = 36 degrees. Since two angles of the triangle are the same, the opposite edges must be the same. The opposite edges are b and c", so the length of b = the length of c". But the length of c = the length of b, so c" = c. This puts the vertex of T at the vertex shared by c and d, causing the extension of s' to intersect that vertex as well. By symmetry of our reflected isosceles triangle (call it S), the extension of the other side of S must intersect that vertex and since the vertex of S is at the intersection of those two sides, the vertex of S is at the vertex of the pentagon.
Now the area of the triangle: The distance from the center of an edge of a pentagon to the opposite vertex will be the sum of the inradius (distance from the edge to the center of the inscribed circle - since the inscribed circle touches at the center of the edge) plus the circumradius (distance from the vertex to the center of the circumscribed circle) or r + R. For any regular polygon, the centers of the inscribed and circumscribed circles will be the same and since the radii of the circles bisect the appropriate angle or side, the center will lie along the line from a vertex of the pentagon to the center of the opposite edge. So we just apply the formula for the area of a triangle: (1/2) * base * height, where a is the base and (r + R) is the height.
