Let $\mathcal{H}$ be the hyperbola with foci at $(\pm 5, 0)$ and vertices at $(\pm 3, 0),$ and let $\mathcal{C}$ be the circle with center $(0,0)$ and radius $4.$ Given that $\mathcal{H}$ and $\mathcal{C}$ intersect at four points, what is the area of the quadrilateral formed by the four points?
For the hyperbola $\mathcal{H},$ we have $a=3$ and $c=5,$ so $b= \sqrt{c^2-a^2} = 4.$ Thus, the hyperbola has equation \[\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1,\]or \[16x^2 - 9y^2 = 144.\]Meanwhile, the equation for the circle is $x^2 + y^2 = 16.$ To find the points of intersection, we solve these two equations simultaneously. Adding $9$ times the second equation to the first equation gives $25x^2 = 288,$ so $x = \pm \frac{12\sqrt2}{5}.$ Then we have \[y^2 = 16 - x^2 = 16 - \frac{288}{25} = \frac{112}{25},\]so $y = \pm \frac{4\sqrt7}{5}.$ Therefore, the four points of intersection form a rectangle with side lengths $\frac{24\sqrt2}{5}$ and $\frac{8\sqrt7}{5},$ so its area is $\frac{24\sqrt2}{5} \cdot \frac{8\sqrt7}{5} = \boxed{\frac{192\sqrt{14}}{25}}.$
[asy]
void axes(real x0, real x1, real y0, real y1)
{
	draw((x0,0)--(x1,0),EndArrow);
    draw((0,y0)--(0,y1),EndArrow);
    label("$x$",(x1,0),E);
    label("$y$",(0,y1),N);
    for (int i=floor(x0)+1; i<x1; ++i)
    	draw((i,.1)--(i,-.1));
    for (int i=floor(y0)+1; i<y1; ++i)
    	draw((.1,i)--(-.1,i));
}
path[] yh(real a, real b, real h, real k, real x0, real x1, bool upper=true, bool lower=true, pen color=black)
{
	real f(real x) { return k + a / b * sqrt(b^2 + (x-h)^2); }
    real g(real x) { return k - a / b * sqrt(b^2 + (x-h)^2); }
    if (upper) { draw(graph(f, x0, x1),color,  Arrows); }
    if (lower) { draw(graph(g, x0, x1),color,  Arrows); }
    path [] arr = {graph(f, x0, x1), graph(g, x0, x1)};
    return arr;
}
void xh(real a, real b, real h, real k, real y0, real y1, bool right=true, bool left=true, pen color=black)
{
	path [] arr = yh(a, b, k, h, y0, y1, false, false);
    if (right) draw(reflect((0,0),(1,1))*arr[0],color,  Arrows);
    if (left) draw(reflect((0,0),(1,1))*arr[1],color,  Arrows);
}
void e(real a, real b, real h, real k)
{
	draw(shift((h,k))*scale(a,b)*unitcircle);
}
size(8cm);
axes(-6,6,-6,6);
xh(3,4,0,0,-5,5);
e(4,4,0,0);
dot((5,0)^^(-5,0)^^(3,0)^^(-3,0));
for (int i=-1; i<=1; i+=2)
	for (int j=-1; j<=1; j+=2)
    	dot((i*12*sqrt(2)/5,j*4*sqrt(7)/5));
draw((-1*12*sqrt(2)/5,-1*4*sqrt(7)/5)--(12*sqrt(2)/5,-1*4*sqrt(7)/5)--(12*sqrt(2)/5,4*sqrt(7)/5)--(-12*sqrt(2)/5,4*sqrt(7)/5)--cycle,dotted);
[/asy]