Four square
A square contains two overlapping squares. What is the total of the shaded regions?
Problem
The square ABCD has an area of 196. It contains two overlapping squares; the larger of these squares has an area 4 times that of the smaller and the area of their overlap is 1.
What is the total area of the shaded regions?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The large square has area $196 = 14^2$, so it has side-length $14$.
The ratio of the areas of the inner squares is $4 : 1$, so the ratio of their side-lengths is $2 : 1$.
Let the side-length of the larger inner square be $2x$, so that of the smaller is $x$.
The figure is symmetric about the diagonal AC and so the overlap of the two inner squares is also a square which therefore has side-length $1$.
Thus the vertical height can be written as $x + 2x - 1$.
Hence $3x - 1 = 14$ and so $x = 5$.
So the small shaded square has an area of $25$ and the large shaded square has an area of $100$.
Therefore, the total shaded area = $25 + 100 - 1 = 124$
Note also that the two unshaded rectangles have side-lengths of $14 - 2x$ and $14 - x $; that is $4$ and $9$.
So the total unshaded area is $36 \times 2 = 72$, and therefore the total shaded area is $196-72=124$