It is exactly that and you even can use the same proof for a triangle, a cone or higher dimensional analogies. You always get 1/dimension. Proof: Take height 1

It is exactly that and you even can use the same proof for a triangle, a cone or higher dimensional analogies. You always get 1/dimension. Proof: Take height 1 for simplicity of notation. Then we get the volume as the integral over crosssections (Fubini). Each cross section is rescaled by a factor (1-z), where z is the height. Rescaling changes the cross sections measure by (1-z)^(n-1). Integrate that to get 1/n. Done.