First, determine the length of the sides of the square. To do so, consider the given diagonal and apply Pythagorean formula.`c^2=a^2+b^2` `8^2=s^2+s^2` `64=2s^2` `64/2=(2s^2)/2` `32=s^2` `sqrt32=s` `4sqrt2=s` Now that the length of the sides is known, we need to take note that the diameter of a circle inscribed in...

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First, determine the length of the sides of the square. To do so, consider the given diagonal and apply Pythagorean formula.

`c^2=a^2+b^2`

`8^2=s^2+s^2`

`64=2s^2`

`64/2=(2s^2)/2`

`32=s^2`

`sqrt32=s`

`4sqrt2=s`

Now that the length of the sides is known, we need to take note that the diameter of a circle inscribed in a square is equal to its side.

Hence, the diameter of the circle is 4sqrt2 ft. And its radius is:

`r = d/2=(4sqrt2)/2`

`r=2sqrt2`

Now that the radius is known, proceed to solve for the circumference and area of the circle.

The formula for circumference and area of circle are:

`C=2pir` and `A= pir^2`

Plug-in the value of r to each formula.

`C=2pi(2sqrt2)=4sqrt2pi`

`A= pi(2sqrt2)^2= pi(4*2)=8pi` **Hence, the circumference of the circle is `4sqrt2pi` ft and its area is `8 pi` square ft.**