The area of a charged rotating (Kerr) black hole is given by \begin{equation} \label{eqn:kerr-area-relation} A = 4\pi \left[ 2 M^2 + 2 (M^4 – L^2 – M^2 Q^2)^{1/2} – Q^2 \right] \end{equation} This relation can be inverted to express the mass $M$ as a function of charge $Q$, area $A$ and angular momentum $L$: \begin{equation} \label{eqn:kerr-mass-relation} M = \left[ \frac{A}{16\pi} + \frac{4\pi L^2}{A} + \frac{Q^2}{2} + \frac{\pi Q^4}{A} \right]^{1/2} \end{equation} As can be seen this is a homogenous function of the variables $(A,L,Q^2)$ of degree $n = 1/2$. From the first law of black hole thermodynamics¹, we know that the change in the mass of the black hole can be written as: \begin{equation} \label{eqn:kerr-first-law} dM = T dA + \Omega dL + \Phi dQ \end{equation} where $ T, \Omega, \Phi$ are the temperature ², angular velocity and electric potential of the black hole respectively. Now, Euler’s homogenous function theorem³ tells us that any function homogenous in its variables of degree $n$, satisfies: \begin{equation} \label{eqn:euler-homogenous} n f(\vec{x}) = \sum_{i=1}^N x_i \frac{\partial f}{\partial x_i} \end{equation} where $\vec{x} = {x_1,x_2,\ldots,x_N} $. \ref{eqn:euler-homogenous} along with \ref{eqn:kerr-mass-relation}, implies that: \begin{eqnarray} \frac{1}{2} M &= A \frac{\partial M}{\partial A} + L \frac{\partial M}{\partial L} + Q^2 \frac{\partial M}{\partial Q^2} \ &= AT + L\Omega + \frac{1}{2} Q \Phi \end{eqnarray} where we have used \ref{eqn:kerr-first-law} to determine the differentials of $M$ with respect to the other variables. In the third term on the r.h.s, we have used: \begin{equation} Q^2 \frac{\partial M}{\partial Q^2} = Q^2 \frac{\partial M}{\partial Q} \frac{\partial Q}{\partial Q^2} = \frac{Q}{2} \frac{\partial M}{\partial Q} \end{equation} This gives us the Smarr formula relating the mass of a charged, rotating black hole to its charge, angular momentum and area: \begin{equation} \label{eqn:smarr-relation} M = 2AT + 2L\Omega + Q\Phi \end{equation}