To begin, we need to find the radius of the circle. The circumference of a circle is given as follows:

\(\displaystyle \small C = 2\pi r\),

where \(\displaystyle r\) is the radius.

Then, a circle with circumference \(\displaystyle 6\pi\) will have the following radius:

\(\displaystyle 6\pi = 2\pi r\)

\(\displaystyle \small \frac{6\pi}{2\pi}=\frac{2\pi r}{2\pi}\)

\(\displaystyle \small 3=r\)

Using the radius, we can now solve for the area:

\(\displaystyle \small A=\pi r^2\)

\(\displaystyle A=\pi(3)^2\)

\(\displaystyle \small A=9\pi\)

The area of the circle is \(\displaystyle \small 9\pi\).

If the diameter is 7, then the radius is half of 7, or 3.5.

Plug this value for the radius into the equation for the area of a circle:

\(\displaystyle A=r^2\pi=(3.5)^2\pi=12.25\pi\)

The orange region is a composite of two figures:

One is a rectangle measuring 20 by 10, which, subsequently, has area

\(\displaystyle 20 \cdot 10 = 200\).

The other is a semicircle with diameter 10, and, subsequently, radius 5. Its area is 

\(\displaystyle \frac{1}{2} \pi \cdot 5^{2} = \frac{25}{2} \pi\).

Add the areas:

\(\displaystyle A = 200 + \frac{25}{2} \pi\)

The formula to find the area of a circle is \(\displaystyle A=\pi\cdot r^2\).

First you must find the radius from the diameter.

\(\displaystyle r=\frac{1}{2}d \rightarrow r=\frac{1}{2}\cdot 10=5\)

In this case it is, 

\(\displaystyle A=5^2 \cdot \pi = 25\cdot 3.14= 78.5\)

The formula for finding the area of a circle is \(\displaystyle \pi r^{2}\). In this formula, \(\displaystyle r\) represents the radius of the circle.  Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius.  In order to do this, we divide the diameter by \(\displaystyle 2\).

\(\displaystyle \frac{15}{2}=7.5\)

Now we use \(\displaystyle 7.5\) for \(\displaystyle r\) in our equation.

\(\displaystyle \pi (7.5)^{2}=176.7146 \: in^{2}\)

First, solve for radius:

\(\displaystyle r=\frac{d}{2}=\frac{6}{2}=3\)

Then, solve for area:

\(\displaystyle A=r^2\pi=3^2\pi=9\pi\)

The area of a circle can be calculated using the formula:

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\) is the diameter of the circle, and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

\(\displaystyle Area=\frac{\pi d^2}{4}=\frac{\pi\times 4^2}{4}=4\pi \Rightarrow Area\approx 4\times 3.14\Rightarrow Area\approx 12.56 \ cm^2\)

The area of a circle can be calculated using the formula:

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\)  is the diameter of the circle and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

\(\displaystyle Area=\frac{\pi (4t)^2}{4}=\frac{16\pi t^2}{4}=4\pi t^2 \Rightarrow Area\approx 4\times 3.14\times t^2\)

\(\displaystyle \Rightarrow Area\approx 12.56t^2\)

Use the formula:

\(\displaystyle \text{Area}=\pi\times r^2\)

Where \(\displaystyle r\) corresponds to the circle's radius.

Since \(\displaystyle r=4\):

\(\displaystyle \text{Area}=\pi\times(4^2)\)

\(\displaystyle \text{Area}=16\pi\)