The radius of a circle with diameter 17 inches is half that, or 8.5 inches. The area of the circle is

From the given figure, you should notice that the base of the triangle is the same as the diameter of the circle.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the triangle.

Recall how to find the area of a circle:

Now recall the relationship between the radius and the diameter.

Plug in the value of the diameter to find the value of the radius.

Now, plug in the value of the radius in to find the area of the circle.

Next, recall how to find the area of a triangle.

The height is already given by the question, and remember that the base is the same as the diameter of the circle.

Plug in these values to find the area of the triangle.

We are now ready to find the area of the shaded region.

Remember to round to decimal places.

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

The relationship between the radius and the area of a circle can be given as

.

We can substitute and solve for :

Double this to get the diameter: , which we round to 11.3.

Use the following formula to find the area of a circle:

For the circle in question, plug in the given radius to find the area.

We know the radius is therefore, the area equation becomes,

.

Recall that when a square root is squared you are left with the number under the square root sign. This happens because when you square a number you are multiplying it by itself. In our case this is,

.

From here we can use the property of multiplication and radicals to rewrite our expression as follows,

and when there are two numbers that are the same under a square root sign you bring out one and the other number and square root sign go away.

The radius of a circle with diameter inches is half that, or inches. The area of the circle is

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

The formula for the area of a circle is:

The radius of the circle is , so we plug it into the formula, as follows:

So the area of the circle is .