To find the height of the four isosceles triangles, draw a right triangle from the vertex to the center of the base to the midpoint of one of the sides. If the sides have length A and the overall height is H, then the Pythagorean Theorem says this right triangle's hypotenuse is sqrt(H^2 + (A/2)^2). The area of each isosceles triangle is then 1/2 * sqrt(H^2 + (A/2)^2) * A, so the overall area is four times this plus the base area, namely
A^2 + 2A sqrt(H^2 + A^2/4)
The time you enjoy wasting is not wasted time. Bertrand Russell