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.)


can someone please help me with this question i posted over again cus my teacher reassign it help me

Answer 1

If you draw the picture, you'll basically have a square wedged tightly inside a circle (the corners of the square will be touching the circle)


So we can use the diagonal of the square to find the diameter of the circle. So we need to find the diagonal of the square.


When they say "use the ratio of 1:1:√2", they mean that if the square had side lengths of 1, then the diagonal will have a length of √2


But the side lengths are actually 6, so multiply every term of 1:1:√2 by 6 to get 6:6:6√2


So the diagonal of the square (with side length of 6") is 6√2. So the diameter is 6√2


Now cut this figure in half to get the radius


\[\frac{6\sqrt{2}}{2}=\frac{6}{2}\sqrt{2}=3\sqrt{2}\]


So the radius is \[r=3\sqrt{2}\]


Now use the formula \[A=\pi r^2\] to find the area of the circle


\[A=\pi r^2\]


\[A=\pi (3\sqrt{2})^2\]


\[A=18\pi\]


So the area of the circle is \[18\pi\]


The area of the square is then A=6*6 = 36


So the area of the portion outside the square but inside the circle is \[18\pi-36\]


Finally, divide this by 4 (because there are 4 individual segments) to get the answer \[\frac{18\pi-36}{4}=\frac{9\pi-18}{2}\]


So the area of one segment, which is the region outside the square but inside the circle, is \[\frac{9\pi-18}{2}\]