What is the area of the composite shape shown with sides 10m, 13m, 5m, and 12m?

The perimeter is the sum of all four sides of the trapezoid.

$$P = 10 \text{ m} + 12 \text{ m} + 5 \text{ m} + 13 \text{ m}$$ $$P = 40 \text{ m}$$

The area of a trapezoid is given by the formula $A = \frac{1}{2} (a+b)h$, where $a$ and $b$ are the lengths of the parallel sides and $h$ is the perpendicular height between them.

$$A = \frac{1}{2} (10 \text{ m} + 5 \text{ m}) \times 12 \text{ m}$$ $$A = \frac{1}{2} (15 \text{ m}) \times 12 \text{ m}$$ $$A = 15 \text{ m} \times 6 \text{ m}$$ $$A = 90 \text{ m}^2$$

The perimeter of the trapezoid is $\boxed{40 \text{ m}}$.
The area of the trapezoid is $\boxed{90 \text{ m}^2}$.