# Math Made Easy! How to Find the Surface Area of a Cylinder

## Geometry Tutorial

**Total Surface Area of a Cylinder**

For high school geometry students that are not really "fans" of the geometry subject, it is problems like finding the surface area of a cylinder that often cause kids to shut their text books and give up or find a geometry tutor.

But, don't panic just yet. Geometry, like many types of math, is often so much easier to understand when broken down into bite-sized pieces. This geometry tutorial will do just that - break down the equation for finding the surface area of a cylinder into easy to understand portions.

Be sure to follow along the cylinder surface area problems and solutions in the **Geometry Help Online** section below, as well as to try out the **Math Made Easy! quiz**.

## Equation for Total Surface Area of a Cylinder

**S.A. = 2**π**r ^{2} + 2**π

**rh**

Where: r is the radius of the cylinder and h is height of the cylinder.

**Before beginning be sure you understand the following geometry tutorials:**

## Use Familiar Objects to Visualize Geometric Shapes

## Math Made Easy! Tip

Admittedly, the formula for the surface area of a cylinder isn't too pretty. So, let's try to break the formula apart into understandable pieces. A good math tip is to try to visualize the geometrical shape with an object with which you are already familiar.

**What objects in your home are cylinders?** I know in my pantry I have a lot of cylinders - better known as canned goods.

Let's examine a can. A can is made up of a top and bottom and a side that curves around. If you could unfold the side of a can it would actually be a rectangle. While I am not going to unfold a can, I can easily unfold the label around it and see that it is a rectangle.

**Now you can visualize the total surface area of a cylinder (can):**

- a can has 2 circles, and
- a can has 1 rectangle

**In other words, you can think of the equation of the total area of a cylinder as:**

S.A. = (2)(area of a circle) + (area of a rectangle)

Therefore, in order to calculate the surface area of a cylinder you need to calcuate the area of a circle (twice) and the area of a rectangle (once).

**Let's look at the total surface area of a cylinder equation again and break it down into easy to understand portions.**

**Area of Cylinder = 2**π**r ^{2}**

*(portion 1)*

**+ 2**π

**rh**

*(portion 2)*

**Portion 1:**The first portion of the cylinder equation has to do with the area of the 2 circles (the top and bottom of the can). Since we know that the area of*one*circle is πr^{2}then the area of*two*circles is 2πr^{2}. So, the first part of the cylinder equation gives us the area of the two circles.**Portion 2:**The second portion of the equation gives us the area of the rectangle that curves around the can (the unfolded label in our canned good example).We know that the area of a rectangle is simply its width (w) times its height (h). So why is the width in the second portion of the equation (**2**π**r)(h**) written as**(2**π**r)?**Again, picture the label. Notice that the width of the rectangle when rolled back around the can is exactly the same thing as the circumference of the can. And the equation for circumference is 2πr. Multiply (2πr) times (h) and you have the area of the rectangle portion of the cylinder.

## Geometry Help Online: Surface Area of Cylinder

Check out three common types of geometry problems for finding the surface area of a cylinder given various measurements.

## #1 Find Surface Area of Cylinder Given the Radius and Height

**Problem: **Find the total surface area of a cylinder with a radius of 5 cm. and a height of 12 cm.

**Solution: **Since we know r = 5 and h=12 substitute 5 in for r and 12 in for h in the cylinder's surface area equation and solve.

- S.A. = (2)π(5)
^{2}+ (2)π(5)(12) - S.A. = (2)(3.14)(25) + (2)(3.14)(5)(12)
- S.A. = 157 + 376.8
- S.A. = 533.8

**Answer: **The surface area of a cylinder with a radius of 5 cm. and a height of 12 cm. is 533.8 cm. squared.

## #2 Find the Surface Area of a Cylinder Given the Diameter and Height

**Problem: **What is the total surface area of a cylinder with a diameter of 4 in. and a height of 10 in.?

**Solution: **Since the diameter is 4 in., we know that the radius is 2 in., since the radius is always 1/2 of the diameter. Plug in 2 for r and 10 for h in the equation for the surface area of a cylinder and solve:

- S.A. = 2π(2)
^{2}+ 2π(2)(10) - S.A. = (2)(3.14)(4) + (2)(3.14)(2)(10)
- S.A. = 25.12 + 125.6
- S.A. = 150.72

**Answer:** The surface area of a cylinder with a diameter of 4 in. and a height of 10 in. is 150.72 in. squared.

## #3 Find the Surface Area of a Cylinder Given the Area of One End and the Height

**Problem: **The area of one end of a cylinder is 28.26 sq. ft. and its height is 10 ft. What is the total surface area of the cylinder?

**Solution: **We know that the area of a circle is πr^{2} and we know that in our example the area of one end of the cylinder (which is a circle) is 28.26 sq. ft. Therefore, substitute 28.26 for πr^{2} in the formula for the area of a cylinder. You can also substitute 10 for h since that is given.

S.A. = (2)(28.26) + 2πr(10)

This problem still cannot be solved since we do not know the radius, r. In order to solve for r we can use the area of a circle equation. We know that the area of the circle in this problem is 28.26 ft. so we can substitute that in for A in the area of a circle formula and then solve for r:

**Area of Circle (solve for r):**- 28.26 = πr
^{2} - 9 = r
^{2}*(divide both sides of the equation by 3.14)* - r = 3
*(take the square root of both sides of the equation)*

Now that we know r = 3 we can substitute that into the area of the cylinder formula along with the other substitutions, as follows:

- S.A. = (2)(28.26) + 2π(3)(10)
- S.A. = (2)(28.26) + (2)(3.14)(3)(10)
- S.A. = 56.52 + 188.4
- S.A. = 244.92

**Answer: **The total surface area of a cylinder whose end has an area of 28.26 sq. ft. and a height of 10 is 244.92 sq. ft**.**

## Do you need more geometry help?

If you have another specific problem you need help with related to the total **surface area of the cylinder** please ask in the comment section below. I'll be glad to help out and may even include your problem in the problem/solution section above.

## Comments

Very nicely done! Would have been cute to use Heinz numberetti (number pasta in a cylinder).

It's been a long time since I've had to work out the area of a cylinder, but this brought it all back. Brilliantly explained, in simple terms.

Great work, well done!!!

What a great way to explain math! It makes such a difference to "see" the problem and also see that usual things at home can be used as help! I will show this hub to my daughter who thinks math is a little difficult! Voted up, useful!

Tina

Great resource for students and math teachers!

Oh how clever, ktrapp! You explained this so well. Thank you!

Voted up / interesting / useful

Its cool that you could break it down like that. I like the can example. When I saw the title of this hub, I realized that I had completely forgotten everything from geometry class, and this reminded me again. Good work.

Wow, did I need you back in my days in school. This was made so simple to follow and I finally got it after 30 years with your hub. Thanks for sharing. Where were you in 72?

great way to explain math.

Like the others..you made this geo class easier than my Teacher Ktrap!

LORD

Ay, yi, yi, k! I could have used your help back in the early 70s when I got a D- in geometry. Thanks for the memories! That's when I discovered that English was more of my thang!

Taking the paper off the can to explain the concept was ingenious. Are you sure you shouldn't be a math teacher. You make it look so easy. Great hub!

Oh my...you're a math genius. I think Math don't like me as I don't like it...Lol---

Still loving your math hubs--they are interesting and a great reference tools for a wide variety of people. And they make me smile. :)

Hi ktrapp! Your title says easy...did I miss something? Haha! Numbers and I do not get along. Fantastic hub though!

This is SO NEAT!! I should share this with my old geometry teacher- this would make for such a fantastic lesson!

Ktrapp,

This hub is a great example of how using creativity helps us transform "complex" things into much simplier understanding! It's a desperately needed attribute for cutting through the noise and confusion in our world today! Bravo!

Voted Up and Awesome!

ktrapp: Congratulaitons on hub of the day. This is truly a great hub... even I did well on the quiz!

When was the last time I came across this formula? Many thanks to you for bringing back all the memories, both good and bad. On top of that, congratulations for the hub of the day!

Congratulations on hub of the day.

Excellent job, very creative, that you actually wrote the nice piece of whole article beginning with a simple math formula. Actually on school days, geometry had been easiest subject for me because i also used similar kind of techniques to remember these kinds of formula. You remind me of those old days. Math students will love this article.

Oh, I WILL be a fan and this WILL be bookmarked for my daughters!!! Not only is this in simple terms but it is wonderfully laid out. You are a gem for sure :)

It lets the grey stuff works overtime!

What a great way to teach geometry - I love hands-on math lessons! I am bookmarking this one for my kids, too - granted, they're only 4 and 6 now, but this will definitely come in handy in a few years!

Congratulations...hub of the day, that s cool!!

Clever use of materials to bring home this concept to kids.

Some 'get it' right away; some do not. For those who do not this will be the way to begin. Some of the 'get it' kids may prefer learning this way as well.

I will share this with my teacher friends!!! Thanks for sharing.

I know I commented a while back, but just wanted to say congrats on your well-deserved Hub of the Day! :)

This is a very creative way to break down geometry. Congrats on being the hub of the day!

I told you this before, I think your Math Made Easy series is brilliant. Congrats on Hub of the Day!

Good job, but I would recommend adding the word "total" before "surface area", as many problems ask for the "lateral surface area" that does not include the area of the two circles.

I bet you are a genus. Good job explaining. Even I managed not to get completely lost.

I repeat: Auuuuuuuuuuuuuuuuuuuuuuuuuuuuuugh!!!

Congratulations on Hub of the Day.

I'm not a math person, and you lost me at "portion 2." My brain hurts. LOL (And RE: William Norman's comment--that was my first thought--what if it's an open-ended cylinder without lids--such as a section of pipe? Then only the "flattened out" cylinder dimensions would apply, eh?) :-)

Very well done explanation, anyway, for those who actually may still be studying the subject. At age 63, however, I've given myself a pardon from doing any more math problems. I'm a writer--I don't need math.

Even if I am a math dummy--I've still voted this as up, interesting and useful--because it will be useful to many!

good information and helpful.

Congrats on hub of the day. This is a brilliant way to explain a complicated area formula!

Excellent Hub! Voted up!

I love it when teachers make things fun.....then they automatically become easier to understand. Great job!

Congraulations!

So cool idea.

Wow!!! Can you come and tutor my kid? lol :) Thanks for the hub ktrapp!!

Wow, you made the solution easier to understand. I'm impressed with your analogy between the can and the label around it.

Well said! Congrats on Hub of the Day!

Ktrapp - as I stated earlier, this was so ingenious. Great hub!

Congratulations on hub of the day!

I wish I would have had you for geometry in high school. You make it simple to understand, breaking it down and then building the interest and skill. The visuals are so helpful...voted up!

Great hub!

Interesting hub -- I liked the aha of realizing that the sides of a cylinder are a rectangle -- but in what situation would knowing the total surface area of a cylinder be of practical use? What I often wonder is which size pizza is cheapest on a per square unit basis?

Very well explained. You have made it very easy. Voted up.

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