Question

A circle with radius 16 centimeters is inscribed in a square.
Which is the area of the shaded region?
A. 804.25 square feet
B. 1024 square feet
C. 822.94 square feet
D. 220.16 square feet

Answer 1

To find the area of the shaded region when a circle with a radius of 16 centimeters is inscribed in a square, we first need to determine the area of both the square and the circle.

1. **Area of the circle:**
The formula for the area of a circle is given by:
\[
A = \pi r^2
\]
where \( r \) is the radius.

Here, the radius \( r = 16 \) cm:
\[
A_{circle} = \pi (16)^2 = \pi \times 256 \approx 804.25 \text{ cm}^2
\]

2. **Area of the square:**
Since the circle is inscribed in the square, the diameter of the circle will be equal to the side length of the square. The diameter \( d \) can be calculated as:
\[
d = 2r = 2 \times 16 = 32 \text{ cm}
\]
Therefore, the side length \( s \) of the square will also be \( 32 \) cm.

The area of the square is given by:
\[
A = s^2
\]
\[
A_{square} = (32)^2 = 1024 \text{ cm}^2
\]

3. **Area of the shaded region:**
The shaded region is the area of the square minus the area of the circle:
\[
A_{shaded} = A_{square} - A_{circle}
\]
\[
A_{shaded} = 1024 - 804.25 \approx 219.75 \text{ cm}^2
\]

Since none of your options match the computed area and due to the units in the options being in square feet, we will convert the area from square centimeters to square feet (1 square foot = 929.0304 square centimeters):
\[
A_{shaded} \approx 219.75 \text{ cm}^2 \times \frac{1 \text{ ft}^2}{929.0304 \text{ cm}^2} \approx 0.236 \text{ ft}^2
\]

Clearly, the options provided seem to be incorrect or based on different assumptions. Please verify the values or units. If it is confirmed to remain in centimeters, the shaded area we calculated is approximately 219.75 cm². If we're looking at options listed, please clarify the question or ensure units are consistent.