Question

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.)

Answer 1

For an inscribed square in a circle, the diagonal of the square is equal to the diameter of the square.
So the diagonal of the square= sqrt(2)*6
Diameter=sqrt(2)*6
Radius=sqrt(2)*3
Now the square divides the circle into 4 sectors of equal area.
So just subtract the area of the square from that of the circle.
This gives
pi*(sqrt(2)*3)^2-6*6
=18pi-36
Now this is the area of all four segments,therefore the area of one segment is
(18pi-36)/4
=4.5pi-9