Three circles are arranged in a row so that each is tangent to the
circles next to it. the radii of the two circles at the two ends are 5
and 3. what is the length of the line segment ab that passes through
the center of each circle ?

πd is the circumference of a circle, so with a diameter of 4 a single circle has a circumference of 4π. 1/3 of each circle is in contact with the belt, so that total part is still 4π. Then, the part of the belt connecting from tangent to tangent is 2 radii, so 3 areas is 12. The belt would then be 12+4π cm.