Let R=sphere radius Let r=cylinder radius then sphere Cap height = R-1 considering that V.Sphere.Cap= 1/3*PI*(Cap.Hight)^2*(3*R-Cap.Hight) => V.Sphere.Cap = 1/3*PI*(R-1)^2*(3R-(R-1)) V.Sphere.Cap = 1/3*PI*(R^2-2R+1)(2R+1) V.Sphere.Cap = 1/3*PI*(2R^3-3R^2+1)

V.Ring=Vol.Sphere -2*V.Sphere.Cape-V.cylender V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^2+1)) - 2*PI*r^2 but r^2=R^2 -1^2=R^2-1 substitute on above we get V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^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

## Let R=sphere radius Let

Let R=sphere radius

Let r=cylinder radius

then sphere Cap height = R-1

considering that V.Sphere.Cap= 1/3*PI*(Cap.Hight)^2*(3*R-Cap.Hight)

=> V.Sphere.Cap = 1/3*PI*(R-1)^2*(3R-(R-1))

V.Sphere.Cap = 1/3*PI*(R^2-2R+1)(2R+1)

V.Sphere.Cap = 1/3*PI*(2R^3-3R^2+1)

V.Ring=Vol.Sphere -2*V.Sphere.Cape-V.cylender

V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^2+1)) - 2*PI*r^2

but r^2=R^2 -1^2=R^2-1

substitute on above we get

V.Ring=4/3*PI*R^3 - 2*(1/3*PI*(2R^3-3R^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