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Let R=sphere radius
Let r=cylinder radius
then sphere Cap height = R1
considering that V.Sphere.Cap= 1/3*PI*(Cap.Hight)^2*(3*RCap.Hight)
=> V.Sphere.Cap = 1/3*PI*(R1)^2*(3R(R1))
V.Sphere.Cap = 1/3*PI*(R^22R+1)(2R+1)
V.Sphere.Cap = 1/3*PI*(2R^33R^2+1)
V.Ring=Vol.Sphere 2*V.Sphere.CapeV.cylender
V.Ring=4/3*PI*R^3  2*(1/3*PI*(2R^33R^2+1))  2*PI*r^2
but r^2=R^2 1^2=R^21
substitute on above we get
V.Ring=4/3*PI*R^3  2*(1/3*PI*(2R^33R^2+1))  2*PI*R^2 + 2*PI
V.Ring=4/3*PI*R^3  4/3*PI*R^3 + 2*PI*R^2 2/3*PI  2*PI*R^2 + 2*PI
V.Ring= 2/3*PI + 2*PI
V.Ring= 4/3 *PI