Express answer in exact form.

Find the area of one segment formed by a square with sides of 6" inscribed in a circle.

(Hint: use the ratio of 1:1:√2 to find the radius of the circle.)

The answer is 5.13 in²

Step 1. Calculate the diameter of the circle (d).

Step 2. Calculate the radius of the circle (r).

Step 3. Calculate the area of the circle (A1).

Step 4. Calculate the area of the square (A2).

Step 5. Calculate the difference between two areas (A1 - A2) and divide it by 4 (because there are total 4 segments) to get the area of one segment formed by a square with sides of 6" inscribed in a circle.

Step 1:

The diameter (d) of the circle is actually the diagonal (D) of the square inscribed in the circle. The diagonal (D) of the square with side a is:

D = a√2 (ratio of 1:1:√2 means side a : side a : diagonal D = 1 : 1 : √2)

If a = 6 in, then D = 6√2 in.

d = D = 6√2 in

Step 2.

The radius (r) of the circle is half of its diameter (d):

r = d/2 = 6√2 / 2 = 3√2 in

Step 3.

The area of the circle (A1) is:

A = π * r²

A = 3.14 * (3√2) ² = 3.14 * 3² * (√2) ² = 3.14 * 9 * 2 = 56.52 in²

Step 4.

The area of the square (A2) is:

A2 = a²

A2 = 6² = 36 in²

Step 5:

(A1 - A2) / 4 = (56.52 - 36) / 4 = 20.52/4 = 5.13 in²