Cylinders - Math

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What is the surface area of a cylinder of height in., with a radius of in?

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Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * $r^2$$

For our problem, this is:

$\pi * $4^2$ = \pi * 16$

You need to double this for the two bases:

$2 * 16 * \pi = 32 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 4 * 15 = 120 * \pi$

Therefore, the total surface area is:

$32\pi + 120\pi = 152\pi$

What is the surface area of a cylinder having a base of radius in and a height of in?

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Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * $r^2$$

For our problem, this is:

$\pi * $11^2$ = \pi * 121$

You need to double this for the two bases:

$2 * 121 * \pi = 242 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 11* 3 = 66 * \pi$

Therefore, the total surface area is:

$242 * \pi + 66 * \pi = 308\pi$

What is the surface area of a cylinder with a height of in. and a diameter of in?

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Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. Notice, however that the diameter is inches. This means that the radius is . Now, the equation for one base is:

$A = \pi * $r^2$$

For our problem, this is:

$\pi * $2^2$ = \pi * 4$

You need to double this for the two bases:

$2 * 4 * \pi = 8 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 2 * 7 = 28 * \pi$

Therefore, the total surface area is:

$28\pi + 8\pi = 36\pi$

The volume of a cylinder with height of is $275\pi$ . What is its surface area?

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To begin, we must solve for the radius of this cylinder. Recall that the equation of for the volume of a cylinder is:

$V = \pi * $r^2$ * h$

For our values this is:

$275\pi = \pi * $r^2$ * 11$

Solving for , we get:

Hence,

Now, recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * $r^2$$

For our problem, this is:

$\pi * $5^2$ = \pi * 25$

You need to double this for the two bases:

$2 * 25 * \pi = 50 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 5 * 11 = 110 * \pi$

Therefore, the total surface area is:

$50\pi + 110\pi = 160\pi$

What is the surface area of a cylinder of height in, with a radius of in?

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Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * $r^2$$

For our problem, this is:

$\pi * $4^2$ = \pi * 16$

You need to double this for the two bases:

$2 * 16 * \pi = 32 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 4 * 15 = 120 * \pi$

Therefore, the total surface area is:

$32\pi + 120\pi = 152\pi$

What is the volume of a cylinder with a radius of 6 meters and a height of 11 meters? Use 3.14 for $\pi$ .

Note: The formula for the volume of a cylinder is:

$V=\pi $r^2h$$

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To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: . Multiply all of these out and you get . The units are cubed because volume is always cubed.

The volume of a cylinder whose height is twice the diameter of its base is one cubic yard. Give its radius in inches.

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The volume of a cylinder with base radius and height is

$V = \pi $r^{2}$h$

The diameter of this circle is ; its height is twice this, or . Therefore, the formula becomes

$V = \pi $r^{2}$ \cdot 4r = 4\pi $r^{3}$$

Set this volume equal to one and solve for :

$4\pi $r^{3}$ = 1$

$4\pi $r^{3}$ \div 4\pi = 1 \div 4\pi$

$r ^{3}= $\frac{1}{4\pi }$$

$r =\sqrt[3]{ $\frac{1}{4\pi }$} =\frac{ \sqrt[3]{ 1 }$} { \sqrt[3]{ 4\pi }} =\frac{ \sqrt[3]{ 1 }$ \cdot \sqrt[3]{ 2 $\pi^{2}$ } } { \sqrt[3]{ 4\pi } \cdot \sqrt[3]{ 2 $\pi^{2}$ }}=\frac{ \sqrt[3]{ 2 $\pi^{2}$$ } } {\sqrt[3]{ $8\pi^{3}$ }}=\frac{ \sqrt[3]{ 2 $\pi^{2}$$ } } {2\pi }$

This is the radius in yards; multiply by 36 to get the radius in inches.

$$\frac{ \sqrt[3]{ 2 $\pi^{2}$$ } } {2\pi } \cdot 36 = $\frac{ 18 \sqrt[3]{ 2 $\pi^{2}$$ } } {\pi }$

What is the volume of a cylinder with a height of in. and a radius of in?

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This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

$V = \pi * $r^2$ * h$

For our values this is:

$A = \pi * $12^2$ * 5=720\pi$

This is the volume of the cylinder.

What is the volume of a cylinder with a height of in. and a radius of in?

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This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

$V = \pi * $r^2$ * h$

For our values this is:

$A = \pi * $2.5^2$ * 14=87.5\pi$

This is the volume of the cylinder.

What is the radius of a cylinder with a volume of $562.5\pi$ and a height of ?

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Recall that the equation of for the volume of a cylinder is:

$V = \pi * $r^2$ * h$

For our values this is:

$562.5\pi = \pi * $r^2$ * 10$

Solve for :

Using a calculator to calculate $\sqrt{56.25}$ , you will see that

What is the height of a cylinder with a volume of $646.875\pi$ and a radius of ?

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Recall that the equation of for the volume of a cylinder is:

$V = \pi * $r^2$ * h$

For our values this is:

$646.875\pi = \pi * $7.5^2$ * h= \pi * 56.25 * h$

Solve for :

What is the surface area of a cylinder with diameter 4 and height 6? The equation to calculate the surface area of a cylinder is:

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

A cylinder has a volume of 16 and a radius of 4. What is its height?

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

Find the surface area of the following right cylinder:

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

Plug in 4 for , and you get $16\pi$ , then multiply by 2 for 2 circles to get $32\pi .$

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

$2\pi r h$

you would get $80\pi .$

To get your final answer, add $32 \pi$ and $80\pi$ to get $112\pi .$

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

and plug in and .

Given a cylinder with radius of 5cm and a height of 10cm, what is the surface area of the entire cylinder?

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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 = $\pi $r^2$$

So the area of the circle = $25\pi$ , and since there are two circles we have

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 = $2\pi r$

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

$10\cdot 10\pi =100\pi$

So the total surface area =

$50\pi +100\pi =150\pi \ $cm^{2}$$

How many gallon cans of paint must be purchased in order to put a single coat of paint over the surface of a cylindrical water tank if the tank is 75 feet high and 25 feet in radius, and each gallon can of paint covers 350 square feet?

Assume that there is a side, a top, and a bottom to be painted.

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First, use the formula $A = 2\pi r (r+h)$ to find the surface area of the tank in feet.

$A = 2\pi r (r+h) =A = 2\pi \cdot 25 \cdot (25+75) \approx 15,708$

Now divide by 350, remembering to round up.

$15,708 \div 350 \approx 44.9$

45 cans of paint need to be purchased.

The circumference of the base of a cylinder is and the height of the cylinder is . What is this cylinder's surface area? Round to the tenths place.

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The formula to find the surface area of a cylinder is $A= 2 \pi rh + 2 \pi $r^2$$ , where is the height of the cylinder and 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 component of the equation, but not for the 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: $C = \pi \cdot d$ , where is diameter. Radius can be calculated by taking half of the diameter. This means that radius and circumference are related in terms of .

Therefore, the first step for this problem is to solve for .

$C = \pi \cdot d$

$6.28 = \pi \cdot d$

$\frac{6.28}{\pi}$=\frac{\pi \cdot d}{\pi}$

$d=1.99\approx2:cm$

Because the diameter is , that means that the radius must be .

Now the surface area can be solved for after the and values are substituted into the equation.

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

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

$A = 24 \pi +2\pi$

$A = 26 \pi$

$A = $81.68:cm^2$ $\approx81.7:cm^2$$

Find the surface area of the cylinder below.

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

$\text{Area of Bases}$=2\pi $r^2$

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

$$\text{Lateral Surface Area}$=2\pi r h$

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

$$\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.

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

Make sure to round to places after the decimal point.

Find the surface area of the given cylinder.

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

$\text{Area of Bases}$=2\pi $r^2$

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

$$\text{Lateral Surface Area}$=2\pi r h$

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

$$\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.

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

Make sure to round to places after the decimal point.

Find the surface area of the given cylinder.

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

$\text{Area of Bases}$=2\pi $r^2$

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

$$\text{Lateral Surface Area}$=2\pi r h$

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

$$\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.

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

Make sure to round to places after the decimal point.