Question

A circle is circumscribed abouta square whose side length is 6 in.Find the area of the circle.

Answer 1

Step-by-step explanation: To solve this question we need to know the formula of the area of the circle as shown below

`A_(circle)=pi*r^2`

As we can see above we need first to know the circle radius which can be expressed in terms of diameter as follows

`r=(d)/(2)`

where

r ---> circle radius

d---> circle diameter

Step 1: We can see we do not have the information about the circle's diameter or radius to be able to calculate the circle's area, BUT, we do have information about the square inside of it and if we look over the picture below

We can see that the circle's diameter is the same as the diagonal of the square, so all we need to do is calculate the square's diagonal to find our diameter using the following formula.

`D=a√(2)`

where

D ----> is the diagonal of the square

a ----> is the side of the square

Step 2: Let's calculate as follows

`\begin{gathered} D=a√(2) \\ D=6√(2) \\ D\cong8.485 \end{gathered}`

It means that the circle's diameter is approximately 8.485 so

`\begin{gathered} r=(d)/(2) \\ r=(8485)/(2) \\ r\cong4.2 \end{gathered}`

Step 3: Now we are able to calculate the area of the circle as follows

`\begin{gathered} A_(c\imaginaryI rcle)=p\imaginaryI *r^2 \\ A_(c\imaginaryI rcle)=p\imaginaryI *4.2^2 \\ A_(c\imaginaryI rcle)=17.64pi\text{ or 55.418} \end{gathered}`

Final answer: So the final answer is approximate 17.64pi or rounded to 18pi