# A square is inscribed in a circle, side of the square is 2*squareroot2. What is the circumference of the circle in terms of pi.

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Given that the side of the square is 2sqrt2.

Then, we know that the diagonal pf the square is the diagonal of the circle.

Let us calculate.

The diagonal = sqrt(side^2 + side^2)

= sqrt( 2sqrt2)^2 + (2sqrt2)^2

= sqrt ( 8 +8) = sqrt 16 = 4

Then, the diagonal of the circle is 4 units.

Now we need to find the circumference is the circle.

We know that the circumference is given by:

C = 2*r *pi

But r = diagonal/2 = 4/2 = 2

==> C = 2*2 * pi = 4pi

**Then, the circumference of the circle is 4*pi units.**

The side of the square is inscribed in a circle is given to be 2sqrt2.

So the diagonal of the square = sqrt{2*square of the side of the square } = sqrt{2* (2*sqrt2)^2} = sqrt{16} = 4.

Therefore the diagonal of the square is 4.

The diagonal of the square must be the diameter of the square inscribed in the circle.

Therefore the radius of the circle = diameter of the circle/2 = 4/2 = 2.

Therefore the circumference of the circle = 2pi*radius = 2pi*2 = 4pi.

We know that the diagonal of the inscribed square is the diameter of the circle.

We'll determine the diagonal applying the Pythagorean theorem.

The diagonal is the hypothenuse of the both right angled triangles formed by it.

We'll note the hypothenuse by a:

a^2 = (2sqrt2)^2 + (2sqrt2)^2

a^2 = 8 + 8

a^2 = 16

a = 4

We'll consider only the positive value of the hypothenuse.

**The circumference of a circle is:**

C = 2pi*r, but 2r = a = diameter

C = a*pi

**C = 4pi**