The diagonal of a square is 8 ft. Find the circumference and tha area of a circle inscribed in the square.

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

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.

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oldnick | (Level 1) Valedictorian

Posted on

L  (square side ) is D `sqrt(2)` /`2`  D diagonal

the half side L is  the radius of the circle we'r seaching for:

`R` `=` `D` `sqrt(2)` /`4`

so the cirumference C is:

`C` `=` `2``pi` `R` =`D` `pi` `sqrt(2)` /`2` `=` `17,710` `ft`

the Area A is:

`A` `=` `pi` `R` `^2` =`pi` `D` `^2` / `8` =`25,12` `ft` `^2`

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