To calculate the area of the shaded region in this figure, we need to break the problem into steps. The diagram shows a quarter-circle with a triangle inside it, where:

1. The radius of the quarter-circle is 28 cm.
2. The base of the triangle is 21 cm.

Step 1: Area of the quarter-circle

The area of a full circle is given by the formula: $A = \pi r^2$ Since this is a quarter-circle, its area is: $A_{\text{quarter-circle}} = \frac{1}{4} \pi r^2$ Substitute $r = 28 \, \text{cm}$: $A_{\text{quarter-circle}} = \frac{1}{4} \pi (28)^2 = \frac{1}{4} \pi (784) = 196 \pi \, \text{cm}^2$

Step 2: Area of the triangle

The area of a triangle is given by: $A = \frac{1}{2} \text{base} \times \text{height}$ Here, the base is $21 \, \text{cm}$ and the height is $28 \, \text{cm}$: $A_{\text{triangle}} = \frac{1}{2} (21)(28) = \frac{1}{2} (588) = 294 \, \text{cm}^2$

Step 3: Area of the shaded region

The shaded region is the area of the quarter-circle minus the area of the triangle: $A_{\text{shaded}} = A_{\text{quarter-circle}} - A_{\text{triangle}}$ Substitute the values: $A_{\text{shaded}} = 196 \pi - 294$ For an approximate numerical result, use $\pi \approx 3.14$: $A_{\text{shaded}} \approx 196 (3.14) - 294 = 615.44 - 294 = 321.44 \, \text{cm}^2$

Final Answer:

The area of the shaded region is approximately: $\boxed{321.44 \, \text{cm}^2}$

Let me know if you would like any additional explanations or help with this! Here are related questions to expand:

1. What is the exact value of the shaded area in terms of $\pi$?
2. How would the solution change if the triangle's dimensions were different?
3. How do you calculate the perimeter of the shaded region?
4. What is the total area if the figure is rotated into a semicircle?
5. Can you calculate the area of the unshaded triangle in another circle's quadrant?

Tip: Always double-check the dimensions for precision when working with diagrams!