Estimate the area of the land to the nearest square meter

Let point A be the origin $(0,0)$.

• A = $(0,0)$
• Q = $(16, -8)$
• P = $(44, -16)$
• B = $(56, 0)$
• S = $(26, 32)$
• R = $(16, 18)$

The formula for the area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ is:

$$\text{Area} = \frac{1}{2} | (x_1y_2 + x_2y_3 + \dots + x_ny_1) - (y_1x_2 + y_2x_3 + \dots + y_nx_1) |$$

We list the coordinates and repeat the first point at the end:
$(x_1, y_1) = (0, 0)$
$(x_2, y_2) = (16, -8)$
$(x_3, y_3) = (44, -16)$
$(x_4, y_4) = (56, 0)$
$(x_5, y_5) = (26, 32)$
$(x_6, y_6) = (16, 18)$
$(x_7, y_7) = (0, 0)$ (repeat of the first point)

$$(0 \times -8) + (16 \times -16) + (44 \times 0) + (56 \times 32) + (26 \times 18) + (16 \times 0)$$ $$= 0 - 256 + 0 + 1792 + 468 + 0$$ $$= 2004$$

$$(0 \times 16) + (-8 \times 44) + (-16 \times 56) + (0 \times 26) + (32 \times 16) + (18 \times 0)$$ $$= 0 - 352 - 896 + 0 + 512 + 0$$ $$= -736$$

$$\text{Area} = \frac{1}{2} | 2004 - (-736) |$$ $$\text{Area} = \frac{1}{2} | 2004 + 736 |$$ $$\text{Area} = \frac{1}{2} | 2740 |$$ $$\text{Area} = 1370$$

The area of the land is 1370 square meters. Since the question asks to estimate to the nearest square meter, the answer is already in the correct format.

The estimated area of the land is $\text{1370 square meters}$.