Surface Area of a Cylinder
Surface Area = (C(h)) + (2π r²)
Where:
C = circumference of the cylinder
h = height of the cylinder
π = a Greek symbol for pi; 3.14159...
r = radius of the cylinder
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A cylinder is a tube-shaped three-dimensional object. Some common cylinder-shaped objects are soda cans, paper towel and wrapping paper tubes, tall candles, Pringles cans, drums, and marshmallows. I say cylinder-shaped because most of these objects are not true cylinders.
Properties of a Cylinder
Cylinders are formed by two sets of parallel lines; two lines that make up the top and the bottom and two lines that make up the sides.
Cylinders are round, with closed circles as their tops and bottoms.
Most people think that all cylinders are taller than they are wide, but this is not always the case. The object below is a just as much a cylinder as the object at the top of this document.
The Nature of the Shape
The easiest way for me to explain a cylinder so that you understand is to have you first think of a rectangle. Now, take the ends of that rectangle (either the ends at the sides or those at the top and bottom) and curve them so that the rectangle forms into a circular ring.
This is the body of the cylinder (also referred to as its lateral area.)
Have this shape stand so that its two open ends are at its top and bottom (as pictured above.) To complete this shape so that it's a true cylinder, its open ends must be closed.
Since both of these ends are circles, it would only make sense for us to close them with circles of the same size. In other words, the ends of our shape must be closed with circles that share the same circumference and area.
The surface area of the cylinder is the area of the rectangle that makes up its body plus the area of the two circles that close the body.
Note: Keep in mind that since cylinders are made from all parallel lines, the two circles that close the top and the bottom are the same size. That means that to get the total surface area of the ends of the cylinder, you must find the universal area of these circles and multiply it by 2.
The Treasure Hunt to a Cylinder's Surface Area
Finding the surface area of a cylinder can seem like a treasure hunt because there are so many steps to follow to finally find the answer. Below is the series of steps and an explanation for each.
Surface Area = Lateral Surface Area + Area of the Ends
Lateral Surface Area = Circumference(height)
Circumference = 2π(radius) -or- π(diameter)
Area of the Ends = 2π(radius)²
Surface Area
The surface area of any object is the sum of all areas of its exposed sides that make up its surface.
Lateral Surface Area
The value of the area of the rectangle that forms the cylinder's body is the value of the cylinder's lateral surface area.
How do we find the value of the rectangle now that it's curved? Well, think about it; to find the area of a rectangle, you simply multiply its length by its width. Therefore, since one of these (the length or the width) has been bent to form a circle, you must use the formula for the circumference of a circle (2πr) to find its value.
After you have that, multiply it by the other side of the rectangle (this is the height of your cylinder) to find the rectangle's area.This will, in turn, give you the lateral surface area of your cylinder.
Circumference
The circumference of a circle is the length of the line that forms it.As was just explained, this is the value of either the width or the length of the rectangle that was curved to form the cylinder.To find this value, one must use the formula 2πr or πd.
r stands for the radius of the cylinder's body and d stands for its diameter. Since the radius of a circle is half of its diameter either one of the formulas (2πr or πd) can be used to reach the same value for circumference.
To find out much more about the circumference of circles (and what their radius and diameters are), click here.
Area of the Ends
For a cylinder to be complete, it must be closed off at both ends. Think of it as a Pringles can that's open on both ends. You don't want all of those heavenly potato chips to spill out, so you must close the bottom and the top.
To find the area of a circle, we use the equation π r². Since we know that these circles that will close the cylinder are the same circumference and area of the ends of the cylinder's body, we can use the same value for r in this equation as we did to find the cylinder's body.
Also, to save some more time, recall that to find the circumference of the cylinder earlier, we used 2πr. For finding the total area of the ends of our cylinder, we use 2πr². The only difference between this formula and the one for the circumference of the cylinder is the squaring of r. Therefore, to find the area of the ends, all you need to do is multiply the solution to 2πr by the value of r.
This will give you the sum of the area of both ends of the cylinder. Now, all you need to do is add this value to the lateral surface area. This addition will complete the sum of the areas of all exposed sides of the cylinder, and therefore, will give the cylinder's surface area.
All About π
Pi was introduced to mathematics only because people were getting too bored while doing equations and wanted to add some foodly fun to the grand scheme of things. They chose pi to be part of a circle's equation because pies are almost always in the shape of a circle.
Okay, so maybe that wasn't completely right, but at least I gave my insight (oh, geez, now I'm even rhyming...)
Pi was found many, many thousands of years ago to be (as David Blatner puts it) "the ratio of a circle's circumference to its diameter." This means that for every perfect circle:
In other words, the circumference of any perfect circle ever made or thought of is larger than its diameter by 3.14159...
Therefore, since that is the unquestionable principle for all circles, when one needs to find a circle's circumference, he/she simply solves that equation for C.
When using pi, don't go through the trouble of finding all the numbers after the decimal point that you can get your hands on. Most people use 3.14 in place of pi to avoid unnecessary trouble.**
The name and symbol for pi came from the Greek alphabet. Pi is the sixteenth letter of this alphabet.
The exact value of pi will never be known because there is an infinite amount of numbers that come after the 3. There are world records for who has calculated the most of these infinite digits.
In 1995, Simon Plouffe found an equation (called the BBP Formula) which allowed him to calculate any digit after the decimal point of pi. He and his colleagues used it to calculate the ten-billionth digit.
(For more information on pi than you probably even care about, check out the book The Joy of Pi, written by David Blatner, or go to its website: http://www.joyofpi.com/)
Example
The radius of an 8-inch-high cylinder is 4 inches. Find the cylinder's total surface area.
Recall the process of finding a cylinder's surface area:
Surface Area = Lateral Surface Area + Area of the Ends
Lateral Surface Area = Circumference(height)
Circumference = 2π(radius) -or- π(diameter)
Area of the Ends = 2π(radius)²
First, let's find the circumference so we can get the lateral surface area:
C = 2πr
Plug in the r value:
C = 2π(4)
Multiply pi by the radius:
3.14 (4) = 12.56
That makes the equation:
C = 2(12.56)
Multiply the two values together:
2(12.56)= 25.12
That means that rounded to the nearest 4th decimal, C = 25.1327 inches
Now that we know the circumference is 25.1327 in, we can go on to find the lateral surface area of the shape:
Lateral Surface Area = C(h)
Plug in the values for C and h:
Lateral Surface Area = (25.12 in)(8in)
Multiply them together:
(25.1327)(8) = 200.96 in²
The lateral surface area of the cylinder is 200.96 square inches.
Now, we must find the area of the ends and add that to the lateral surface area. Recall the formula for area of the ends (the sum of the area of the cylinder's two circles:)
Area of Ends = 2πr²
Remember that we already found 2πr in the previous step (this is the value for the circumference of the cylinder.) Therefore, all that needs to be done is multiplying that by r:
2πr = 25.12 in²
r = 4in
25.12(4) = 100.48 in²
I'll also find the area of the ends the standard way (not "cheating" by using a number I already found beforehand) just in case you have to do this for a test and you have to show each and every step of your work.
Area of a circle = πr²
Total Area of Cylinder's Ends (two circles with the same area) = 2πr²
According to Order of Operations,
we know to square r first:
r = 4 in
(4)² = 16
That gives:
2π(16)
Multiply pi by 16:
3.14(16) = 50.24
Now our equation is:
2(50.24)
Multiply 50.24 by 2:
2(50.24) = 100.48 in²
So:
Lateral Surface Area = 200.96in²
Area of Ends = 100.48 in²
We now need to add these two values together:
200.96+ 100.48 = 301.44 in²
The total surface area of our cylinder is 301.44 square inches.
A Little Bit About Square Units
Note that I kept track of the units and instead of inches, my answer was in square inches (square in or in²). When calculating the surface area of a cylinder, the units of your answer will always be squared (square feet, square inches, etc.) This is because when assigning one number for the measurement of the different sides of a cylinder, both the length and the width of each side need to be accounted for.
By using square units you say how many single unit squares fit into the larger shape being measured. So, for example, when I say the surface area of a cylinder is 150 sq. in. it means that 150 smaller squares with the area of 1 sq in. fit into the total area of the surface of the cylinder. This way, both the length and the width of the square is included in the answer.
** Our Calculators Like to Show Off
Although π (pi) has an infinite amount of numbers after the decimal point, people commonly use 3.14 as its value when solving an equation that uses it. Our calculator, however, uses a more accurate version of pi with thirteen digits after the decimal point (3.1415926535898.) This means that if the same values that I use for the example in this help file are plugged into our calculator, the answer will be slightly different (and more accurate.)
So why do I use a less accurate number if it gives a less accurate answer? I use the rounded version because that is what most teachers will have you use for pi in school. In some cases, you may get counted off for giving a different answer than your math book (or your teacher) has.
Also, it is much easier to learn and remember the rounded value. Not to mention, if I used the number with thirteen decimals in each example, it would take much longer to work through and the long string of digits after the decimal might very well become confusing.
Works Cited
Blatner, David. "The Joy of π." The Joy of π. 1998. David Blatner. 19 June 2006.
<http://www.joyofpi.com/thebook.html>.
Weisstein, Eric W. "BBP Formula." MathWorld. 17 January 2004. Wolfram Research,
Inc. 20 June 2006. <http://mathworld.wolfram.com/BBPFormula.html>.
"13.2: Cylinders." The Geometry Center. 5 February 1999. The University of
Minnesota. 7 July 2006.
<http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node57.html>.
&"Square Inch." Wikipedia.com. 17 May 2006. Wikimedia Foundation, Inc. 13 June
2006. <http://en.wikipedia.org/wiki/Square_inch>.
"Surface Area and Volume." Integrated Publishing. 1998. Integrated Publishing. 7 July
2006. <http://www.tpub.com/math1/19d.htm>.
"Surface Area Formulas." math.com. 2005. Math.com. 13 June 2006.
<http://www.math.com/tables/geometry/surfareas.htm>.
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