What if you were given a solid three-dimensional figure with congruent enclosed circular bases that are in parallel planes? How could you determine how much two-dimensional and three-dimensional space that figure occupies? After completing this Concept, you'll be able to find the surface area and volume of a cylinder.

### Watch This

### Guidance

A **cylinder** is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.

A cylinder has a ** radius** and a

*height.*

A cylinder can also be ** oblique** (slanted) like the one below.

##### Surface Area

**Surface area** is the sum of the area of the faces of a solid. The basic unit of area is the square unit.

**Surface Area of a Right Cylinder:** \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}.

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

To see an animation of the surface area, click http://www.rkm.com.au/ANIMATIONS/animation-Cylinder-Surface-Area-Derivation.html, by Russell Knightley.

##### Volume

To find the **volume** of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit. For cylinders, volume is the area of the circular base times the height.

**Volume of a Cylinder:** \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.

#### Example A

Find the surface area of the cylinder.

\begin{align*}r = 4\end{align*} and \begin{align*}h = 12\end{align*}.

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

#### Example B

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.

We need to solve for the radius, using the circumference.

\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 \ units^2\end{align*}

#### Example C

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*}

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### Guided Practice

1. Find the volume of the cylinder.

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?

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.

**Answers:**

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

2. Solve for \begin{align*}r\end{align*}.

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

3. We need to solve for the radius, using the circumference.

\begin{align*}2 \pi r &=80 \pi\\ r&=40\end{align*}

Now, we can find the surface area.

\begin{align*}SA &=2 \pi (40)^2+(80 \pi )(36)\\ &= 3200 \pi +2880 \pi\\ &= 6080 \pi \ units^2\end{align*}

### Explore More

- Two cylinders have the same surface area. Do they have the same volume? How do you know?
- A cylinder has \begin{align*}r = h\end{align*} and the radius is 4 cm. What is the volume?
- A cylinder has a volume of \begin{align*}486 \pi \ ft.^3\end{align*}. If the height is 6 ft., what is the diameter?

- A right cylinder has a 7 cm radius and a height of 18 cm. Find the volume.

Find the volume of the following solids. Round your answers to the nearest hundredth.

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

- \begin{align*}V=6144 \pi \ units^3\end{align*}
- 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.
- The circumference of the base of a cylinder is \begin{align*}34 \pi \ cm\end{align*} and the height is 20 cm. Find the total surface area.
- The lateral surface area of a cylinder is \begin{align*}30 \pi \ m^2\end{align*} and the height is 5m. What is the radius?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.4.