\(\displaystyle C = \pi d\)

\(\displaystyle 36 \pi = \pi d\)

\(\displaystyle d = 36\)

The perimeter of a circle = 2 πr = πd

Therefore d = 36

The first step is to sketch two circles touching at a single point. In order to maximize the length of \(\displaystyle XY\), the point \(\displaystyle X\) and the point \(\displaystyle Y\) would need to be on opposite ends, as shown in the diagram. If the radius of a circle is \(\displaystyle 6\), then the diameter would be \(\displaystyle 2r = 2(6) = 12\). Therefore, the length of \(\displaystyle XY\) would be \(\displaystyle 2d = 2(12) = 24\).

The diameter is the maximum distance between two points on a circle's perimeter. The diameter passes through the circle's center.

First solve for the radius:

\(\displaystyle A=\pi r^{2} = 9\pi\) 

\(\displaystyle r = 3 \; in\)  

Note that \(\displaystyle d = 2r\), where \(\displaystyle r\) is the radius and \(\displaystyle d\) is the diameter. 

Therefore, \(\displaystyle d = 6\; in\).

The area of a circle is given by the equation \(\displaystyle A = \pi r^2\), where \(\displaystyle A\) is the area and \(\displaystyle r\) is the radius. Use the given area in this equation and solve for \(\displaystyle r\) to find the circle's radius.

\(\displaystyle 60\pi = \pi r^2\)

\(\displaystyle \frac{(60\pi)}{\pi} = \frac{(\pi r^2)}{\pi}\)

\(\displaystyle 60 = r^2\)

\(\displaystyle \sqrt{60} = \sqrt{r^2}\)

\(\displaystyle r = \sqrt{60} = 2\sqrt{15}\)

To find the circle's diameter, multiply its radius by \(\displaystyle 2\). 

\(\displaystyle 2\sqrt{15} * 2 = 4\sqrt{15}\)\(\displaystyle cm\)

The diameter of a circle can be written as \(\displaystyle d=2r\), where \(\displaystyle r\) is the radius and \(\displaystyle d\) is the diameter. 

\(\displaystyle (7) 2 = 14\)

Therefore the diameter of the circle is 14 inches.

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr².

\(\displaystyle A=\pi (5/2)^2=6.25\pi\)

Before we can find the diameter of this circle, we need to find its radius. We need to use the formula for the area of a cirlce:

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

Given that the area is \(\displaystyle 9\pi\), we can find the radius

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

cancel the pi and then square root it to find 'r'.

\(\displaystyle 9=r^2\)

\(\displaystyle 3=r\)

Now that the radius is found, we can find the diamater by multiplying it by 2.

\(\displaystyle D=2r=2(3)=6\)

Recall how to find the area of a circle:

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

Next, plug in the information given by the question.

\(\displaystyle 144\pi=\pi\times\text{radius}^2\)

From this, we can see that we can solve for the radius.

\(\displaystyle \text{radius}^2=144\)

\(\displaystyle \text{radius}=\sqrt{144}=12\)

Now recall the relationship between the radius and the diameter.

\(\displaystyle \text{diameter}=2\times\text{radius}\)

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

\(\displaystyle \text{diameter}=2\times 12 = 24\)