Problem (Geometry):
The perimeter of equilateral $\triangle P Q R$ is 12. The perimeter of regular hexagon $S T U V W X$ is also 12. What is the ratio of the area of $\triangle P Q R$ to the area of $S T U V W X$ ?

Solution:
Since the hexagon has perimeter 12 and has 6 sides, then each side has length 2 .

Since equilateral $\triangle P Q R$ has perimeter 12 , then its side length is 4 .

Consider equilateral triangles with side length 2.

Six of these triangles can be combined to form a regular hexagon with side length 2 and four of these can be combined to form an equilateral triangle with side length 4 .


Note that the six equilateral triangles around the centre of the hexagon give a total central angle of $6 \cdot 60^{\circ}=360^{\circ}$ (a complete circle) and the three equilateral triangles along each side of the large equilateral triangle make a straight angle of $180^{\circ}\left(\right.$ since $3 \cdot 60^{\circ}=180^{\circ}$ ). Also, the length of each side of the hexagon is 2 and the measure of each internal angle is $120^{\circ}$, which means that the hexagon is regular. Similarly, the triangle is equilateral.

Since the triangle is made from four identical smaller triangles and the hexagon is made from six of these smaller triangles, the ratio of the area of the triangle to the hexagon is $4: 6$ which is equivalent to $2: 3$.

Final answer: $\frac{2}{3}$