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# Surface Area and Volume of Cylinders

## Surface area and volume of solids with congruent circular bases in parallel planes.

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Cylinders

What if you wanted to figure out how much paper you needed for the label of a can? How could you use the surface area of a cylinder to help you?

### Cylinders

A cylinder is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed. Just like a circle, the cylinder has a radius for each of the circular bases. Also, like a prism, a cylinder can be oblique, like the one to the right.

##### Surface Area

Surface area is the sum of the areas of the faces. Let’s find the net of a right cylinder. One way for you to do this is to take a label off of a soup can or can of vegetables. When you take this label off, we see that it is a rectangle where the height is the height of the cylinder and the base is the circumference of the base. This rectangle and the two circular bases make up the net of a cylinder.

From the net, we can see that the surface area of a right cylinder is

\begin{align*}& \ \underbrace{2 \pi r^2} \qquad + \qquad \underbrace{2 \pi r} h\\ &\text{area of} \qquad \qquad \ \text{length}\\ & \ \text{both} \qquad \qquad \quad \ \ \text{of}\\ &\text{circles} \qquad \qquad \text{rectangle}\end{align*}

Surface Area of a Right Cylinder: If \begin{align*}r\end{align*} is the radius of the base and \begin{align*}h\end{align*} is the height of the cylinder, then the surface area is \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}.

##### Volume

Volume is the measure of how much space a three-dimensional figure occupies. The basic unit of volume is the cubic unit: cubic centimeter \begin{align*}(cm^3)\end{align*}, cubic inch \begin{align*}(in^3)\end{align*}, cubic meter \begin{align*}(m^3)\end{align*}, cubic foot \begin{align*}(ft^3)\end{align*}, etc. Each basic cubic unit has a measure of one for each: length, width, and height. The volume of a cylinder is \begin{align*}V=(\pi r^2)h\end{align*}, where \begin{align*}\pi r^2\end{align*} is the area of the base.

Volume of a Cylinder: If the height of a cylinder is \begin{align*}h\end{align*} and the radius is \begin{align*}r\end{align*}, then the volume would be \begin{align*}V=\pi r^2 h\end{align*}.

If an oblique cylinder has the same base area and height as another cylinder, then it will have the same volume. This is due to Cavalieri’s Principle, which states that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

#### Finding the Surface Area

1. Find the surface area of the cylinder.

\begin{align*}r = 4\end{align*} and \begin{align*}h = 12\end{align*}. Plug these into the formula.

\begin{align*}SA & = 2 \pi (4)^2 + 2 \pi (4)(12)\\ & = 32 \pi + 96 \pi\\ & = 128 \pi\end{align*}

2. The circumference of the base of a cylinder is \begin{align*}16 \pi\end{align*} and the height is 21. Find the surface area of the cylinder.

If the circumference of the base is \begin{align*}16 \pi\end{align*}, then we can solve for the radius.

\begin{align*}2 \pi r& = 16 \pi\\ r & =8\end{align*}

Now, we can find the surface area.

\begin{align*}SA& = 2 \pi (8)^2+(16 \pi)(21)\\ & = 128 \pi + 336 \pi\\ & = 464 \pi\end{align*}

#### Finding the Volume

Find the volume of the cylinder.

If the diameter is 16, then the radius is 8.

\begin{align*}V=\pi 8^2 (21)=1344 \pi \ units^3\end{align*}

### Examples

#### Example 1

Find the volume of the cylinder.

\begin{align*}V=\pi 6^2 (15)=540\pi \ units^3\end{align*}

#### Example 2

If the volume of a cylinder is \begin{align*}484 \pi \ in^3\end{align*} and the height is 4 in, what is the radius?

Substitute what you know to the volume formula and solve for \begin{align*}r\end{align*}.

\begin{align*}484 \pi & = \pi r^2 (4)\\ 121& = r^2\\ 11 & = r\end{align*}

#### Example 3

Find the volume of the solid below.

This solid is a parallelogram-based prism with a cylinder cut out of the middle. To find the volume, we need to find the volume of the prism and then subtract the volume of the cylinder.

\begin{align*}V_{prism} & =(25 \cdot 25)30=18750 \ cm^3\\ V_{cylinder} & = \pi (4)^2 (30)=480\pi \ cm^3\end{align*}

The total volume is \begin{align*}18750 - 480\pi \approx 17242.04 \ cm^3\end{align*}.

### Review

1. The lateral surface area of a cylinder is what shape? What is the area of this shape?
2. A right cylinder has a 7 cm radius and a height of 18 cm. Find the surface area and volume.

Find the surface area and volume of the following solids. Leave answers in terms of \begin{align*}\pi\end{align*}.

Find the value of \begin{align*}x\end{align*}, given the surface area.

1. \begin{align*}SA = 1536 \pi \ units^2\end{align*}
2. The area of the base of a cylinder is \begin{align*}25 \pi \ in^2\end{align*} and the height is 6 in. Find the lateral surface area.
3. The circumference of the base of a cylinder is \begin{align*}80 \pi \ cm\end{align*} and the height is 36 cm. Find the total surface area.
4. The lateral surface area of a cylinder is \begin{align*}30 \pi \ m^2\end{align*}. What is one possibility for height of the cylinder?
5. Charlie started a business canning artichokes. His cans are 5 in tall and have diameter 4 in. If the label must cover the entire lateral surface of the can and the ends must overlap by at least one inch, what are the dimensions and area of the label?
6. Find an expression for the surface area of a cylinder in which the ratio of the height to the diameter is 2:1. If \begin{align*}x\end{align*} is the diameter, use your expression to find \begin{align*}x\end{align*} if the surface area is \begin{align*}160\pi\end{align*}.
7. Two cylinders have the same surface area. Do they have the same volume? How do you know?
8. A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
9. A cylinder has a volume of \begin{align*}486 \pi \ ft.^3\end{align*}. If the height is 6 ft., what is the diameter?
10. The area of the base of a cylinder is \begin{align*}49 \pi \ in^2\end{align*} and the height is 6 in. Find the volume.
11. The circumference of the base of a cylinder is \begin{align*}80 \pi \ cm\end{align*} and the height is 15 cm. Find the volume.
12. The lateral surface area of a cylinder is \begin{align*}30\pi \ m^2\end{align*} and the circumference is \begin{align*}10\pi \ m\end{align*}. What is the volume of the cylinder?

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### Vocabulary Language: English

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Cavalieri's Principle

States that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

Oblique Cylinder

An oblique cylinder is a cylinder with bases that are not directly above one another.