The problem asks to determine the area of the shaded region in a circle. Given the image:

• The circle has a radius $r = 14$ cm.
• $\angle AOB$ is a right angle, which means it is $90^\circ$.

Steps to solve:

1. Calculate the area of the sector $AOB$:

   The area of a sector is given by: $\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$ Here, $\theta = 90^\circ$ and $r = 14$ cm. $\text{Area of sector} = \frac{90^\circ}{360^\circ} \times \pi \times 14^2 = \frac{1}{4} \times \pi \times 196 \text{ cm}^2 = 49\pi \text{ cm}^2$

2. Calculate the area of triangle $OAB$:

   Since $\triangle OAB$ is a right triangle with both legs as the radii of the circle: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 14 \times 14 = 98 \text{ cm}^2$

3. Determine the area of the shaded region:

   The shaded region is the area of the sector minus the area of the triangle: $\text{Area of shaded region} = 49\pi \text{ cm}^2 - 98 \text{ cm}^2$

Approximate Numerical Value:

To give a numerical approximation, use $\pi \approx 3.14$: $\text{Area of shaded region} = 49 \times 3.14 - 98 = 153.86 \text{ cm}^2 - 98 \text{ cm}^2 = 55.86 \text{ cm}^2$

Final Answer:

The area of the shaded region is approximately 55.86 cm².

Would you like any further details or clarifications?

Related Questions:

1. How would the result change if the radius were doubled?
2. What is the general formula for the area of a shaded region for any given central angle $\theta$?
3. How does the area of the shaded region compare if $\theta$ is 180° instead of 90°?
4. What is the circumference of the circle in this problem?
5. How would you calculate the perimeter of the shaded region?

Tip: Remember to always check the angle measure and ensure it's correctly applied in sector area calculations.