To find the area of a polygon, we can use the shoelace formula (or Gauss's area formula) for polygons, which is effective when the vertices of the polygon are given.

For a polygon with vertices $(x_1, y_1)$, $(x_2, y_2)$, ..., $(x_n, y_n)$, the area $A$ is given by:

$A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|$

For the triangle with vertices $P(1,1)$, $Q(-2,1)$, and $R(-1,-4)$, let's apply the formula:

1. $P(1, 1)$
2. $Q(-2, 1)$
3. $R(-1, -4)$

Now apply the coordinates to the formula:

$A = \frac{1}{2} \left| (1 \times 1 + (-2) \times (-4) + (-1) \times 1) - (1 \times (-2) + 1 \times (-1) + (-4) \times 1) \right|$

Let's compute that step by step.