Question

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

(sorry for the bad diagram, but it's a square inside the circle. I've used tick marks to indicate that the sides of the square are equal)

Answer 2

we can draw a right triangle inside the square. the hypotenuse of this right triangle is the radius of the square. additionally, each one of the legs is half the length of the square, or in other words, each leg is 6/2 = 3"

Answer 3

like the problem states: use the ratio 1:1:√2

the triangle we draw is a 45:45:90 degree triangle (if you need me to prove this, I'll gladly do so)

therefore, the ratio of the leg:hypotenuse is 1:1:√2

therefore, since each leg is 3", the hypotenuse is 3*sqrt(2), and thus the radius of the circle is 3*sqrt(2)

Answer 4

last step:

each "segment" made by the circle and square is 1/4 the area in between the circle and the square (shaded like so)



the area in between the circle and the square is just area of the circle - area of the square

or as a formula

pi * r^2 - s^2

where r is the radius of the circle from the previous step, and s is the length of the square's sides (6)

plug in and evaluate. then divide by 4 to get the area of 1 segment.