How can I do this?

Question

Given the circle inscribed in the square with a diagonal of 6 square root 2 in,

1) determine the area of RSTU

2) Determine the area of center O, (O is the center of that circle).

3) Find the shaded region.

Answer 1

You have to use special right triangles for this one. More specifically, a 45 45 90 triangle.

Since the hypotenuse of the triangle is 6√2, then the leg of the triangle is 6 (again, this uses special right triangles). Since the leg of the triangle is also a side of the square, the area of the square will be 6x6 = 36 units².

To find the area of the circle, you must know the radius. Since the circle is inscribed in the square, that means the diameter of the circle is 6, which makes the radius 3. The formula for the area of the circle is πr² so π(3²) = 9π.

And finally, without looking at the picture, I don't know where the shaded region is.

Answer 2

I'm guessing a picture like this:

Let each side of the square be "x". Using the triangle SUT, we can apply the Pythagorean Theorem and say x² + x² = (6√2)²

2x² = 36•2

x² = 36•2/2

x² = 36

x = √36

x = 6

So the area of the square is 6•6 = 36 square inches

The area of a circle is πr² and in this case, we will need to do some thinking to get "r". Notice that "r" is half of the side length of the square? So r = 3 since the side length is 6 (from above).

So A_circle = π(3)² = 9π

The area of the shaded portion (if I got that correct in my picture) is A_square - A_circle = 36 - 9π.square inches You can get a decimal approximation by plugging this into a calculator, In doing so I got A ≈ 7.726 sq in