Question

Express answer in exact form.

Answer 1

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

Answer 2

Answer:

4.5 pi in.^2 - 9 in.^2

Step-by-step explanation:

First, bisect the chord (one side of the square) in half. Draw a line from the middle of the chord to the center of the circle. This will bisect the 90o (o = degree sign) central angle in half, creating a 45o angle.  

So, you have a 45-45-90 triangle. Use the rules for 45-45-90 triangles to determine that the hypotenuse is 3 sqrt 2.  

The hypotenuse of the triangle is the radius of the circle. So now you can find out the area of the circle: A = pi * r^2 --> A = 18 pi in.^2.  

From this information, you can figure out the area of the sector. Do 90o (since the sector angle is 90o) divided by 360o: 90/360. You should get 1/4 (this shows that the sector is a quarter of the square). Next, do 1/4 times the area of the circle --> 1/4 * 18 pi = 4.5 pi in.^2.  

Finally, find out the area of the triangle by using the formula for the area of triangles: A = 1/2 * b * h ---> A = 1/2 * 6 * 3 (we know the base and height from earlier calculations with the 45-45-90 triangle). So --> A = 9 in.^2.  

Lastly, to find the area of one segment, do the area of the sector - the area of the triangle: 4.5 pi in.^2 - 9 in.^2 (This is in exact form. The approximate form would be 5.14 in.^2).

Hope this helps! :)