Chapter 12.2, Problem 10MRE

(a)

To determine

To find: The perimeter of the square.

Answer to Problem 10MRE

The perimeter of the square is equal to $32$ .

Explanation of Solution

Given information: It is given that the square is inscribed in a circle with diameter $8\sqrt{2}$ .

Calculation:

Draw a diagram of a square inscribed in a circle as shown below.

  

The diagonal of the square will be equal to diameter of the circle. So, the diagonal of the square is equal to $8\sqrt{2}$ .

The edge of the square can be calculated as:

  $\begin{array}{c}a=d\times \cos 45°\\ =8\sqrt{2}\times \frac{1}{\sqrt{2}}\\ =8\end{array}$

Thus, the side of the square is equal to $8$ .

The perimeter of the square is:

  $\begin{array}{c}P=4a\\ =4\left(8\right)\\ =32\end{array}$

Therefore, the perimeter of the square is equal to $32$ .

(b)

To determine

To find: The circumference of the circle.

Answer to Problem 10MRE

The circumference of the circle is $8\sqrt{2}\pi$ .

Explanation of Solution

Given information: It is given that the square is inscribed in a circle with diameter $8\sqrt{2}$ .

Calculation:

The radius of the circle is:

  $\begin{array}{c}r=\frac{d}{2}\\ =\frac{8\sqrt{2}}{2}\\ =4\sqrt{2}\end{array}$

The formula for the circumference of the circle is given by,

  $C=2\pi r$

Substitute $4\sqrt{2}$ for $r$ in the above formula.

  $\begin{array}{c}C=2\pi \left(4\sqrt{2}\right)\\ =8\sqrt{2}\pi \end{array}$

Therefore, the circumference of the circle is $8\sqrt{2}\pi$ .