Cylinders - Intermediate Geometry

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Question

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

Answer

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

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Question

Find the surface area of the following right cylinder:

Answer

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 .

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Question

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

Answer

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 =

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

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Question

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.

Answer

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.

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Question

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:

$SA=2r^2\pi+2rh\pi$

Answer

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

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

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Question

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.

Answer

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.

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

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

$A = 24 \pi +2\pi$

$A = 26 \pi$

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

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Question

Find the surface area of the cylinder below.

Answer

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.

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Question

Find the surface area of the given cylinder.

Answer

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.

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Question

Find the surface area of the given cylinder.

Answer

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.

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Question

Find the surface area of the given cylinder.

Answer

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.

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

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

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

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(7)^2+2\pi(7)(14)=294\pi=923.63$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(7)^2+2\pi(7)(20)=378\pi=1187.52$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(20)=682\pi=2142.57$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(23)=748\pi=2349.91$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(25)=792\pi=2488.14$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(2)^2+2\pi(2)(15)=68\pi=213.63$

Make sure to round to places after the decimal point.

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Question

Find the surface area of the given cylinder.

Answer

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.

$\text{Surface Area of Cylinder}=2\pi(3)^2+2\pi(3)(16)=114\pi=358.14$

Make sure to round to places after the decimal point.

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Question

The above diagram shows a sphere inscribed inside a cylinder.

The sphere has a surface area of 100. Give the surface area of the cylinder.

Answer

Let be the radius of the sphere. Then the radius of the base of the cylinder is also , and the height of the cylinder is .

The surface area of the cylinder is

$A_{1} = 2\pi r (r+h)$

which, after substituting, is

$A_{1} = 2\pi r (r+2r)$

$A_{1} =2 \pi r (3r)$

$A_{1} =2\cdot 3 \cdot \pi r \cdot r$

$A_{1} = 6 \pi r^{2}$

The surface area of the sphere is

$A_{2} = 4 \pi r^{2}$

Therefore, the ratio of the former to the latter is

$\frac{A_{1} }{A_{2}}= \frac{6 \pi r^{2}}{4\pi r^{2}} = \frac{6}{4} = \frac{3}{2}$

and

$A_{1} = \frac{3}{2} A_{2}$

That is, the cylinder has surface area $\frac{3}{2}$ that of the sphere. Therefore, this area is

$A_{1} = \frac{3}{2} \cdot 100 = 150$ .

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Question

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Answer

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

$A=2(\pi r^2)+(2\pi r)(h)$

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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