Question

find the area of the shaded region, given the perimeter measurements.

Explanation:

Step1: Find area of the rectangle

The area of a rectangle is length × width.
$\text{Area}_{\text{rectangle}} = 66 \times 44 = 2904 \, \text{cm}^2$

Step2: Find area of the trapezoid

The area of a trapezoid is $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$. First, calculate the height of the trapezoid using the Pythagorean theorem: $h = \sqrt{15^2 - 10^2} = \sqrt{225 - 100} = \sqrt{125} = 5\sqrt{5} \, \text{cm}$.
$\text{Area}_{\text{trapezoid}} = \frac{1}{2} \times (48 + 22) \times 5\sqrt{5} = \frac{1}{2} \times 70 \times 5\sqrt{5} = 175\sqrt{5} \approx 390.3 \, \text{cm}^2$

Step3: Subtract trapezoid area from rectangle

The shaded area is the rectangle area minus the trapezoid area.
$\text{Area}_{\text{shaded}} = 2904 - 175\sqrt{5}$

Answer:

$2904 - 175\sqrt{5} \, \text{cm}^2$ (or approximately $2513.7 \, \text{cm}^2$)