In the diagram, square $A B C D$ is made up of 36 squares, each with side length 1 . The area of the square $K L M N$, in square units, is
(A) 12
(B) 16
(C) 18
(D) 20
(E) 25

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-123.jpg?height=295&width=277&top_left_y=2086&top_left_x=1355)
Solution 1

We can find the area of square $K L M N$ by subtracting the areas of the four triangles $K A N$, $N D M, M C L$ and $L B K$ from the area of square $A B C D$.

To do this, we note that square $A B C D$ has a side length of 6 , and that each of the four triangles is right-angled and has one leg of length 2 and the other of length 4. (A "leg" of right-angled triangle is a side that is not the hypotenuse.)

So the area of square $K L M N$ is $6^{2}-4\left[\frac{1}{2}(2)(4)\right]=36-4[4]=20$ square units.

## Solution 2

Since $K L M N$ is a square, its area is the square of its side length; in particular, the area is equal to $N M^{2}$.

To calculate its side length, we look at right-angled triangle $D N M$ and calculate $N M^{2}$ using Pythagoras:

$$
\begin{aligned}
& N M^{2}=N D^{2}+D M^{2} \\
& N M^{2}=2^{2}+4^{2} \\
& N M^{2}=20
\end{aligned}
$$

Therefore, the area of the square is 20 square units.

ANSWER: (D)