The most efficient way of enclosing the circles is first to arrange them so that each circle is tangent to the other two, draw an isosceles triangle around them, and then draw a fourth side, tangent to larger circle, but parallel to the shorter side of the isosceles triangle.

(The other candidate shape is placing the circles in a row and building a longer slimmer trapezoid around them, but for a quick sanity check, that shape would have to be greater in area than 196^2 + 144^2 + 144^2, which is 79888, and as we shall see the shape below is smaller).

Using coordinate geometry I ascertained the dimensions of this trapezoid. I won’t bore you with the messy details, especially as there’s doubtless a far more elegant way to do it. In any case the base measures 336, the top 147, the sloping sides 337.5, the height 324 and therefore the area is 78246 square units.