Set up a coordinate system so that the origin is at the center of the barbell, x-axis horizontal, y-axis vertical. The vertices of the hyperbola are at (-2, 0) and (2, 0). Further, the foci are at (-3, 0) and (3, 0) because the lie on vertical lines connecting the ends of radii of the hemispheres. For the hyperbola [tex]\frac{x^2}{a^2}- \frac{y^2}{b^2}= 1, the distance from the center to each focus is given by so and .

The equation of the hyperbola is .

At the ends of the hyperbolic section, x= 3 and -3, the same as for the foci which are on those lines. At those points . Solve for y. The length of the hyperbolic section is 2y because of the symmetry, and the entire length is that plus the two hemispheric radii: 2y+ 6.