Question

A circle is inscribed in a square with side lengths of 7 inches.
Which expression represents the area of the square not taken up by the inscribed circle?

Answer 1

To find the area of the square not taken up by the inscribed circle, we will first calculate the area of both the square and the circle, and then subtract the area of the circle from the area of the square.

1. **Calculate the area of the square**:
The area \( A_{square} \) of a square with side length \( s \) is given by the formula:
\[
A_{square} = s^2
\]
For a square with side lengths of 7 inches:
\[
A_{square} = 7^2 = 49 \text{ square inches}
\]

2. **Calculate the area of the inscribed circle**:
The radius \( r \) of the inscribed circle is half the side length of the square:
\[
r = \frac{s}{2} = \frac{7}{2} = 3.5 \text{ inches}
\]
The area \( A_{circle} \) of the circle is given by the formula:
\[
A_{circle} = \pi r^2
\]
Substituting the radius:
\[
A_{circle} = \pi (3.5)^2 = \pi \times 12.25 = 12.25\pi \text{ square inches}
\]

3. **Calculate the area not taken up by the circle**:
The area of the square not occupied by the circle is obtained by subtracting the area of the circle from the area of the square:
\[
A_{remaining} = A_{square} - A_{circle} = 49 - 12.25\pi
\]

Thus, the expression that represents the area of the square not taken up by the inscribed circle is:
\[
49 - 12.25\pi
\]