  In this paper we introduce a new formalism for $K$-theory, called squares
$K$-theory. This formalism allows us to simultaneously generalize the usual
three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow
B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B
\leftarrow C$, by defining $K_0$ of exact and subtractive categories to satisfy
a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with
these corners. Examples that rely on this formalism are $K$-theory of smooth
manifolds of a fixed dimension and $K$-theory of (smooth and) complete
varieties. Another application we give of this theory is the construction of a
derived motivic measure taking value in the $K$-theory of homotopy sheaves.
