Since the radius is 4, the area of the base is \(\displaystyle 16\pi\).  To cancel out the \(\displaystyle \pi\), the height must be \(\displaystyle \frac{1}{\pi }\).

If the diameter of the cylinder is 4, the radius is equal to 2. Therefore:

\(\displaystyle SA=2r^2\pi+2rh\pi=2(2^2)\pi+2(2)(6)\pi=8\pi+24\pi=32\pi\)

The answer is \(\displaystyle 112\pi\ in^{2}\).  To find the surface area we need to find the area of the top, bottom, and the round side respectively.  To find the areas of the top and bottom circles, you would need to use the formula \(\displaystyle \pi r^{2}\) . 

Plug in 4 for \(\displaystyle r\), and you get \(\displaystyle 16\pi\), then multiply by 2 for 2 circles to get \(\displaystyle 32\pi .\)

Then to get the area of the round side, you would take the circumference times the height.  Thus with the formula

\(\displaystyle 2\pi r h\) 

you would get \(\displaystyle 80\pi .\) 

To get your final answer, add \(\displaystyle 32 \pi\) and \(\displaystyle 80\pi\) to get \(\displaystyle 112\pi .\)  

Also you could remember the formula \(\displaystyle SA \ cylinder = 2\left (\pi r^{2}\right )+ h\left (2\pi r \right )\)

 and plug in \(\displaystyle 4=r\) and \(\displaystyle 10=h\). 

The surface area of the whole cylinder = (2 * area of circle) + lateral area

Think of the lateral area as the paper label on a can; It wraps around the outside of the can while leaving the top and bottom untouched. The area of the circle, times 2, is to account for the top and the bottom of the cylinder.

Area of a circle = \(\displaystyle \pi r^2\)

So the area of the circle = \(\displaystyle 25\pi\)\(\displaystyle cm^2\), and since there are two circles we have

\(\displaystyle 50cm^2\)

Now for the lateral area. Notice how if we have a can with a paper label, we can take the label, cut it, and unroll it from the can. In this way, our label now looks like a rectangle with a

height = height and the

width = circumference of the circle.

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

So our rectangle is going to have a height of 10 and a width of 10\(\displaystyle \pi\). So the lateral area =

\(\displaystyle 10\cdot 10\pi =100\pi\)

So the total surface area =

\(\displaystyle 50\pi +100\pi =150\pi \ cm^{2}\)

The formula to find the surface area of a cylinder is \(\displaystyle A= 2 \pi rh + 2 \pi r^2\), where \(\displaystyle h\) is the height of the cylinder and \(\displaystyle r\) is the radius. 

In this kind of equation-based problem, it's helpful to ask "What information do I have?" and "What information is missing that I need?" 

The problem provides information for the \(\displaystyle h\) component of the equation, but not for the \(\displaystyle r\) component. Instead, we're given information about the circumference of the circular base of the cylinder. The question that arises now is how the radius can be calculated from the circumference. The formula for circumference is: , where \(\displaystyle d\) is diameter. Radius can be calculated by taking half of the diameter. This means that radius and circumference are related in terms of \(\displaystyle d\). 

Therefore, the first step for this problem is to solve for \(\displaystyle d\).

\(\displaystyle C = \pi \cdot d\)

\(\displaystyle 6.28 = \pi \cdot d\)

\(\displaystyle \frac{6.28}{\pi}=\frac{\pi \cdot d}{\pi}\)

\(\displaystyle d=1.99\approx2\:cm\)

Because the diameter is \(\displaystyle 2\:cm\), that means that the radius must be \(\displaystyle 1\:cm\).

Now the surface area can be solved for after the \(\displaystyle h\) and \(\displaystyle r\) values are substituted into the equation. 

\(\displaystyle Area = 2 \pi (1) (12) + 2 \pi (1)^2\)

\(\displaystyle A= 2 \pi (12)+ 2 \pi (1)\)

\(\displaystyle A = 24 \pi +2\pi\)

\(\displaystyle A = 26 \pi\)

\(\displaystyle A = 81.68\:cm^2 \approx81.7\:cm^2\)

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

\(\displaystyle \text{Lateral Surface Area}=2\pi r h\)

Add the two together to get the equation to find the surface area of a cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi r^2+2\pi r h\)

Plug in the given height and radius to find the surface area.

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(12)^2+2\pi(12)(20)=768\pi=2412.74\)

Make sure to round to \(\displaystyle 2\) places after the decimal point.

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

\(\displaystyle \text{Lateral Surface Area}=2\pi r h\)

Add the two together to get the equation to find the surface area of a cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi r^2+2\pi r h\)

Plug in the given height and radius to find the surface area.

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(1)^2+2\pi(1)(5)=12\pi=37.70\)

Make sure to round to \(\displaystyle 2\) places after the decimal point.

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

\(\displaystyle \text{Lateral Surface Area}=2\pi r h\)

Add the two together to get the equation to find the surface area of a cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi r^2+2\pi r h\)

Plug in the given height and radius to find the surface area.

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(2)^2+2\pi(2)(5)=28\pi=87.96\)

Make sure to round to \(\displaystyle 2\) places after the decimal point.

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

\(\displaystyle \text{Lateral Surface Area}=2\pi r h\)

Add the two together to get the equation to find the surface area of a cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi r^2+2\pi r h\)

Plug in the given height and radius to find the surface area.

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(3)^2+2\pi(3)(8)=66\pi=207.35\)

Make sure to round to \(\displaystyle 2\) places after the decimal point.

To find the surface area of the cylinder, first find the areas of the bases:

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

Next, find the lateral surface area, which is a rectangle:

\(\displaystyle \text{Lateral Surface Area}=2\pi r h\)

Add the two together to get the equation to find the surface area of a cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi r^2+2\pi r h\)

Plug in the given height and radius to find the surface area.

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(4)^2+2\pi(4)(8)=96\pi=301.59\)

Make sure to round to \(\displaystyle 2\) places after the decimal point.