In convex integral geometry and geometric measure theory, Steiner's formula is the name of the following elegant result:
Let $B_n$ be the unit ball in $\mathbf R^n$. If $S$ is a nonempty bounded convex subset of $\mathbf R^n$, then for $0<r<\infty$, \begin{align}\mathscr L^n(S+rB_n)=\sum_{m=0}^n\omega_{n-m}\zeta^m(S)r^{n-m}\end{align} where $\mathscr L^n$ is the Lebesgue measure on $\mathbf R^n$, $S+rB_n$ is the Minkowski sum $\{x:\operatorname{dist}(x,S)<r\}$, and $\omega_k$ denotes the ($k$-dimensional) volume of the $k$-dimensional unit ball. The quantities $\zeta^m(S)$ are called the $m$-th intrisic volume of $S$.
In context of geometric measure theory, an explicit formula for $\zeta^m(S)$ is also given (Federer, Geometric Measure Theory, Theorem 3.2.35).
Let $\theta^*_{n,m}$ be the invariant measure on the space $\mathbf O^*(n,m)$ of orthogonal projections $\mathbf R^n\to\mathbf R^m$. Then \begin{align}\zeta^m(S)=\frac{1}{\beta(n,m)}\int_{\mathbf O^*(n,m)}\mathscr L^m(p(S))d\theta^*_{n,m}(p)\end{align} where $\beta(n,m)$ is a suitable normalization constant.
Despite its innocent looking, Steiner's formula is very technical to prove (a fact reflected by the late appearance in Federer's rather abstruse book). I wonder what was actually proved by Steiner, who is a synthetic geometer in 19th century, and how did he come up with a proof?
