An integral geometry problem is considered on a family of n-dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in Rn+1$\mathbb {R} <^>{n+1}$. More precisely, the reconstruction of a function f(x,y)$f(\mathbf {x,}y)$, x is an element of Rn$\mathbf {x}\in \mathbb {R} <^>{n}$, y is an element of R$y\in \mathbb {R}$, from the integrals of the form f(x,y)dx$f(\mathbf {x,}y) d\mathbf {x}$ extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection-type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of n=1$n=1$ and n=2$n=2$ are provided.