Area. One figure lies inside another

If figure $A$ is entirely contained within figure $B$ (that is, every point of $A$ is also a point of $B$), then the area of $A$ is less than or equal to the area of $B$: $\text{If } A \subseteq B, \text{ then } \operatorname{Area}(A) \leq \operatorname{Area}(B).$

Important properties

• If $A$ is strictly inside $B$ (no part of $A$ touches the boundary of $B$), then $\operatorname{Area}(A) < \operatorname{Area}(B)$.

• If $A$ and $B$ have the same area and $A \subseteq B$, then $A = B$ (they are the same region).

• This property is used to compare areas and prove inequalities in geometry.