Given task: Two $10 \times 10$ cm square napkins were thrown on the table, as shown in the figure. They covered an area of the table equal to $172$ cm$^2$. What is their overlap area?
Answer is: $2 \times 10 \times 10 - 172 =28$ cm$^2$.
Why the solution is exactly that?
2 Answers
$\begingroup$Each napkin has area $100\ \text{cm}^2.$ If when you put them on the table, they do not overlap at all, then they would cover $200\ \text{cm}^2$ of the table. If they overlapped completely, then they would cover only $100\ \text{cm}^2$ of the table. The only way to reduce the area of the table covered by the napkins from $200\ \text{cm}^2$ is for the napkins to overlap. Every $\text{cm}^2$ of area covered by the napkins that's less than $200\ \text{cm}^2$ is because of overlap. So to find the area of their overlap, you must compute the difference between the total area ($200\ \text{cm}^2$) and the area of table that they cover ($172\ \text{cm}^2$).
$\endgroup$ 1 $\begingroup$Say $A$ is the set of points in the first square , $B$ is the set of points in the second square, then
$$Area(A \cup B) = Area(A) + Area(B) - Area(A \cap B) \iff$$$$ Area(A \cap B) = Area(A) + Area(B) - Area(A \cup B) = 100+100 -172 = 28$$
This is similar to the formula $|A \cup B| = |A| + |B| - |A \cap B|$ but with areas instead.
$\endgroup$ 1
