To compute the area of a polygon given its vertices, we can use the Shoelace formula (or surveyor's formula). The formula for the area \( A \) of a polygon with vertices \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\) is given by:
\[ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \]
where \((x_{n+1}, y_{n+1})\) is \((x_1, y_1)\).
Given the coordinates of the vertices:
- \((-3, -1)\)
- \((0, -3)\)
- \((4, 3)\)
- \((1, 5)\)
Let's denote these points as:
- Point 1: \( (x_1, y_1) = (-3, -1) \)
- Point 2: \( (x_2, y_2) = (0, -3) \)
- Point 3: \( (x_3, y_3) = (4, 3) \)
- Point 4: \( (x_4, y_4) = (1, 5) \)
We'll also add the first point at the end to close the polygon:
- Point 5: \( (x_5, y_5) = (-3, -1) \)
Next, we set up the terms for the Shoelace formula:
\[ \begin{array}{cc} x & y \ -3 & -1 \ 0 & -3 \ 4 & 3 \ 1 & 5 \ -3 & -1 \ \end{array} \]
Now, we compute the sums \( \sum (x_i y_{i+1}) \) and \( \sum (y_i x_{i+1}) \):
-
\( x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_5 \):
- \( (-3)(-3) = 9 \)
- \( (0)(3) = 0 \)
- \( (4)(5) = 20 \)
- \( (1)(-1) = -1 \)
Summing these gives: \[ 9 + 0 + 20 - 1 = 28 \]
-
\( y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_5 \):
- \( (-1)(0) = 0 \)
- \( (-3)(4) = -12 \)
- \( (3)(1) = 3 \)
- \( (5)(-3) = -15 \)
Summing these gives: \[ 0 - 12 + 3 - 15 = -24 \]
Now we can apply the Shoelace formula:
\[ A = \frac{1}{2} \left| 28 - (-24) \right| = \frac{1}{2} \left| 28 + 24 \right| = \frac{1}{2} \cdot 52 = 26 \]
Thus, the area of the polygon formed by the given coordinates is \( \boxed{26} \).