Given prime ideals $\mathfrak p_A,\mathfrak p_B$ of commutative rings $A,B$ respectively, to determine $ht(\mathfrak p_A \times \mathfrak p_B)$

Question

Let $A,B$ be commutative rings with unity. Let $\mathfrak p_A,\mathfrak p_B$ be prime ideals of $A$ and $B$ respectively. Is there a way to determine $ht(\mathfrak p_A \times \mathfrak p_B)$ where $\mathfrak p_A \times \mathfrak p_B$ is the product ideal in $A \times B$ in terms of $ht(\mathfrak p_A), ht(\mathfrak p_B)$ ?